UC-NRLF 


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Jones^  Mathematical  Text-Books* 


PRICE    LIST,    JANUARY 


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i2mo,  paper,  \6  pp.     Single  copies  by  mail,  lo  cents. 
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Single  copies  of  these  books  are  sent  free  to  teachers  of  mathematics  for 
inspection.  For  the  most  part  they  follow  well-worn  lines;  but  in  some 
things  there  are  radical  departures;  anil  teachers  are  advised  neither  to 
accept  them  nor  to  reject  them  without  careful  examination.  They  are 
good  books  for  private  reading. 

(JKOIKJE  \V.  JONES,  PubllHhcr, 
No  AcKNTs.  Ithaca,  N.  Y. 


drill-book: 


i^ 


ALG-EBEA 


BY 


Peof.  GEOKGE  "WILLIAM  JONES 


OF 


CORXELL  UNIVERSITY. 


"Sixth'  Edb  to?/!.' '  • 


ITHACA,  N.  Y.: 
GEORGE  W.  JONES. 

1906 


Entered  according  to  Act  of  Congiess,  in  the  year  1892,  by 

GEORGE  WILLIAM  JONES, 
in  the  Office  of  the  Librarian  of  Congress,  at  Washington, 


PREFACE. 


So  far  as  concerns  mathematical  studies  there  are  always  two 
classes  of  pupils :  those  that  will  use  mathematics  later  in 
life  as  one  of  the  tools  of  their  trade — the  engineers,  architects, 
accountants,  teachers,  scientists;  those  that  will  not  so  use  it. 

To  both  of  these  classes  the  careful  study  of  the  elements 
of  geometry  and  algebra  is  important :  to  the  first  class  as  lay- 
ing a  sure  foundation  for  work  in  the  higher  mathematics  and 
for  its  professional  applications;  to  the  others  as  giving  the 
power  and  the  habit  of  exact  statement  and  rigorous  proof. 
In  this  later  day  we  pride  ourselves  on  our  laboratories,  in 
which  the  pupil  comes  face  to  face  with  the  facts  and  forces 
of  nature — every  mathematical  recitation-room  under  an  able 
teacher  is  a  laboratory  in  logic,  and  for  sound  logic  there  is 
always  an  unlimited  demand. 

This  book  is  for  use  by  the  more  advanced  classes  in  the 
high  schools  and  academies  and  the  lower  classes  in  the  col- 
leges; and  its  primary  object  is  to  teach  young  men  and 
women  to  think.  From  the  beginning  the  philosophy  of  the 
subject  is  made  prominent;  and  in  writing  it  the  author  set 
himself  the  double  task  of  writing  a  book  whose  definitions 
should  be  precise  and  whose  proofs  should  be  rigorous,  and  of 
writing  one  so  simple  that  any  diligent  pujoil  could  read  it 
easily. 

But  he  has  not  confined  himself  to  definitions  and  proofs : 
a  large  collection  of  questions  and  exercises  has  been  added; 
and  for  a  good  understanding  of  the  fundamental  principles, 
and  readiness  in  their  use,  quite  as  much  reliance  is  placed  on 

260875 


IV  PREFACE. 

the  questions  as  on  the  text.  It  is  hoped  that  by  his  effort  to 
answer  these  questions  the  pupil  will  be  early  taught  to  think 
earnestly,  to  think  independently,  and  to  think  hard. 

Believing  that  the  elements  of  plane  geometry  are  at  least 
as  simple  as  those  of  algebra,  the  author  has  assumed  some 
knowledge  of  that  subject  in  pupils  using  this  book,  and  he 
has  not  hesitated  to  use  geometric  illustrations  where  their 
greater  concreteness  seemed  to  give  greater  clearness. 

This  book  was  undertaken  as  an  abridgment  of  Oliver, 
Wait,  and  Jones'  Treatise  on  Algebra;  and  at  first  it  was 
hoped  that  cutting  out  the  more  abstruse  portions  of  the  text 
and  the  harder  examples  would  fit  the  larger  work  for  the  use 
of  the  preparatory  schools;  but  after  that  excision  many  al- 
terations were  found  necessary,  and  in  the  end  the  order  of 
topics  was  changed,  new  lines  of  proof  were  adopted,  new 
questions  and  examples  were  prepared,  and  the  whole  book 
was  rewritten. 

The  author  is  indebted  both  to  the  writers  on  algebra,  from 
whose  works  he  has  drawn  freely,  and  to  his  associates  at 
Cornell  University,  who  have  been  unsjiaring  in  their  kindly 
assistance.  In  particular  he  returns  thanks  to  Professor 
Hathaway,  who  outlined  the  discussion  of  tlie  combinatory 
properties  of  the  simple  arithmetic  02ierations,  that  of  meas- 
ures and  multiples,  and  that  of  incommensurable  numbers; 
to  Mr.  John  11.  Tanner,  who  made  the  selections  from  the 
Treatise  and  prepared  the  first  draft  of  the  copy;  to  Miss  Ida 
M.  Metcalf,  wlio  spent  half  a  year  in  giving  form  to  the  text 
and  preparing  tlie  questions  and  exercises;  and  to  Professors 
Oliver  and  McMahon,  who  have  read  the  greater  part  of  the 
book  either  in  mamiscript  or  in  proof. 

But  with  all  tlie  care  he  could  exercise,  he  is  conscious 
that  many  errors  have  crept  in,  and  that  there  arc  many  de- 
fects tliat  only  use  in  the  class-room  can  bring  to  liglit.  He 
will  esteem  it  a  great  favor,  tlu'reforo,  if  liis  fellow-teachers 
will  tell  him  freely  what  they  find  wrong  either  in  method  or 
matter,  in  general  plan  or  detail. 


PREFACE.  V 

SUGGESTIONS   TO   TEACHERS. 

The  author's  aim  lias  been  to  discuss  subjects  in  the  order  of 
their  logical  dependence,  so  as  to  construct  a  continuous  and 
irrefutable  line  of  argument  throughout;  but  it  must  be  re- 
membered that  the  logical  order  is  not  always  the  easiest  or 
the  most  natural,  and  that  not  only  are  the  claims  of  mathe- 
matical science  to  be  satisfied,  but  the  wants  of  the  individual 
immature  pupil  are  to  be  met.  It  may  often  happen,  there- 
fore, that  deviation  from  the  order  of  the  book  will  be  of 
advantage;  and  a  subsequent  review  in  the  order  laid  down 
will  then  show  the  drift  of  the  thought  and  the  links  in  the 
chain  of  reasoning. 

This  point  should  be  early  and  often  impressed  upon  the 
pupil :  that  many  simple  theorems  are  stated  and  proved,  not 
that  he  may  be  convinced  of  their  truth,  for  that  conviction 
can  be  reached  by  repeated  experiment,  but  that  a  firm  foun- 
dation may  be  laid  for  a  logical  structure  of  ever-increasing 
height  and  complexity.  That  is,  they  are  not  ends  in  them- 
selves, but  only  useful  tools  for  future  work. 

If  parts  of  the  text  seem  too  abstruse  for  his  pupils,  or  the 
questions  too  hard,  the  wise  teacher  will  reserve  such  parts 
for  a  later  reading,  and  he  will  choose  for  himself  the  order 
in  which  the  topics  shall  be  ^^resented.  For  example,  he  may 
find  it  well  to  take  up  parts  of  the  second  chapter  before  fin- 
ishing the  first,  or  to  set  parallel  lessons  from  the  two  chapters. 
Theory  and  practice  may  thus  go  hand  in  hand. 

In  this  book  general  principles  are  stated  formally,  as  in 
text-books  on  geometry,  and  illustrations  and  applications 
follow;  but  the  living  teacher  may  well  reverse  this  order, 
and  before  setting  a  new  topic  in  the  book  he  may  draw  out 
the  whole  matter  from  the  pupils'  own  mind,  by  careful  ques- 
tioning after  the  Socratic  method,  first  in  simple  illustrations 
and  then  in  general  principles.  Afterwards  the  pupil  may 
read  and  explain  the  text,  and  answer  the  questions  set  down 
for  exercises.  The  author  recognizes  the  distinction  between 
the  office  of  the  text-book  and  that  of  the  teacher,  and  he 


VI  PREFACE. 

places  the  man  above  the  book;  but  he  has  been  taught  by 
his  ow]i  experience  that  a  book  is  very  usefuh  And  what 
should  the  book  contain  ?  A  treatise  on  any  subject  contains 
the  whole  body  of  knowledge  on  that  subject,  well  digested 
and  well  arranged  and  indexed,  so  that  the  reader  may  find 
all  that  he  seeks  within  its  pages;  but  it  need  contain  no 
exercises  for  pupils  and  no  questions.  A  drill-book  is  more 
modest  in  its  aims:  it  leaves  out  all  that  is  not  necessary  to 
the  main  purpose;  it  i^resents  the  great  principles  in  due  order 
and  in  simi^le  language,  so  that  the  pupil  may  read  them  again 
and  again,  and  it  sets  him,  under  his  teacher's  guidance,  to 
find  out  their  applications ;  it  suggests  to  him  the  best  methods 
of  work;  it  offers  him  lists  of  questions  on  Avliich  he  may  taj>k 
himself,  and  grow  strong  in  the  exercise;  it  helps  the  teacher 
to  cross-examine  him;  it  serves  as  a  standard  to  both  teacher 
and  pupil;  and  it  saves  them  endless  labor  in  the  giving  and 
taking  of  notes  and  in  the  preparation  and  copying  of  exercises. 

It  has  been  the  author's  good  fortune  to  have  a  few  bright 
young  people  come  to  him  every  summer  for  a  more  complete 
preparation  in  elementary  mathematics,  and  he  has  thus  kept 
fresh  in  mind  the  wants  of  beginners.  He  has  found  ihese 
pujiils  needing  a  regular  and  persistent  drill  both  in  the  state- 
ment and  proof  of  the  fundamental  principles  and  in  tlieir 
application;  and  he  has  written  this  book  to  meet  their  wauts. 
He  submits  it  respectfully  to  the  judgment  of  his  fellow- 
teachers,  only  asking  that  they  neither  adopt  it  nor  reject  it 
without  a  very  careful  examination;  for  while  in  most  things 
it  follows  well-worn  lines,  in  others  it  makes  radical  depart- 
ures from  the  common  usage. 

An  ansvver-ljook  (not  a  key)  is  in  preparation  for  the  con- 
venience of  teachers;  and  the  whole  list  of  questions  has  been 
printed  on  cards  for  use  in  the  class-room. 

George  W.  Jones. 

Ithaca,  N.  Y.,  May  3,  1893. 


CONTENTS. 


I.     THE    PRIMARY   OPERATIONS  OF  ARITHMETIC 

SECTION  PAGE 

1.  Namljer, ,  3 

2.  Multiplication  and  division, 6 

3.  Positive  and  negative  numbers, =     .     .  18 

4.  Addition  and  subtraction, .....  22 

0.  Involution  and  evolution, 30 

6.     Questions  for  review,        34 

II.     THE  PRIMARY  OPERATIONS  OF  ALGEBRA. 

1.  Algebraic  expressions, 36 

2.  Addition  and  subtraction, 40 

3.  Multiplication, 40 

4.  Division, 06 

5.  Fractions, 64 

6.  Questions  for  review,        68 

III.     SIMPLE  EQUATIONS. 

1.  One  unknown  element, 72 

2.  Two  unknown  elements, 78 

3.  Three  or  more  unknown  elements, 88 

4.  Questions  for  review,         94 

IV.     MEASURES  AND  MULTIPLES. 

1.  Integers, 100 

2.  Entire  functions  of  one  letter, 108 

3.  Entire  factors, 118 

4.  The  highest  common  measure 128 

5.  The  lowest  common  multiple, 132 

6.  Questions  for  review,        134 


viii  CONTENTS. 

V.     VARIATIONS,  PROPORTION,  INEQUALITIES,  AND 
INCOMMENSURABLE  NUMBERS. 

SECTION  PAGE 

1.  Variation, 136 

2.  Proportion, 142 

3.  Inequalities, 148 

4.  Incommensurable  numbers, 152 

5.  Questions  for  review,        160 

VI.     POWERS  AND  ROOTS. 

1.  The  binomial  theorem, 162 

2.  Fraction  powers,      .     .     .     , .  164 

3.  Radicals, 170 

4.  Roots  of  polynomials,  .     .     .     .     c     .     . 180 

5.  Roots  of  numerals,       186 

6.  Roots  of  binomial  surds, ,     , 192 

7.  Questions  for  review,        196 

VII.     QUADRATIC  EQUATIONS. 

1.  One  unknown  element, 198 

2.  Two  unknown  elements, 206 

3.  Three  or  more  unknown  elements, 212 

4.  Questions  for  review,         214 

VIII.     THE  THREE  PROGRESSIONS,  INCOMMENSURABLE 
POWERS  AND  LOGARITHMS. 

1.  Arithmetic  progression,     ,     ,     .     . 218 

2.  Geometric  progression, 222 

3.  Harmonic  progression, 226 

4.  Incommensurable  powers, 228 

5.  Logarithms, ,     .     o 232 

6.  Questions  for  review,        , 246 

IX.     PERMUTATIONS,  COMBINATIONS  AND  PROBABILITIES. 

1.  Permutations, 250 

2.  Combinations <>     o     <,     .  252 

3.  Probabilities 258 

4.  Questions  for  review -,«.„.  266 


PRIMARY    NOTIONS   FOR  YOUNGER   PUPILS. 


While  there  are  many  new  things  to  be  learned  in  algebra, 
there  is  nothing  contradictory  to  what  has  been  already  learned 
in  arithmetic,  and  there  are  many  points  of  resemblance  be- 
tween these  two  sciences. 

For  example,  the  same  signs  of  operation,  of  grouping,  and 
of  equality,  are  used,  +,  — ,  x,  :,  (),  =;  and  fractions, 
powers,  and  roots  have  the  same  meaning,  and  are  written  in 
the  same  form. 

The  differences  come  largely  from  the  frequent 

USE    OF   LETTERS   TO   REPRESENT   NUMBERS. 

But  there  is  nothing  arbitrary  or  mysterious  about  this 
use;  the  letters,  for  the  most  part,  are  abbreviations  for  words, 
and  sometimes  they  serve  to  make  the  statements  more  general 
than  if  numerals  alone  were  used. 

The  reasoning  is  the  same  whether  the  numbers  be  expressed 
in  words,  in  letters,  or  in  figures. 

For  example,  if  n  stand  for  a  certain  number,  say  a  man's 
age,  or  the  number  of  books  in  his  library;  then  2n  stands  for 
tlie  double  of  this  number,  3;^  for  its  triple,  \7i  for  its  half. 

To  make  this  clearer,  the  pupil  may  answer  these 

QUESTIONS. 

1.  What  is  the  meaning  of  4;^  ?  of  6;^?  of  %n  ?  of  3;i-i?i? 
of     ^(3ji  +  9;i-7;0?     of     3|/(r^i  +  8;^)?     of     n  :  ^n"^ 

If  n  stand  for  GO,  what  are  the  values  of  these  expressions? 

2.  So,  if  X,  y,  z  stand  for  three  numbers,  say  the  cost  of  an 
algebra,  a  reader,  and  a  grammar,  for  what  does  x-\-y  stand  ? 
x  +  y  +  z'^  x  +  y-z'^  bx  +  2y-^z'^   ^{2x  +  y-z)'^    5x-4y-z? 

3.  If  a;  =  50,  ^  =  30,  z  =  70,  find  the  values  of  the  ex- 
pressions written  above,  and  of  these: 

,  y      x  +  y      X  ^  X  X      z-^x       z 

z  z    '      z      ^'      z  +  y'  y'        y    '      y 

ix 


X  PRIMARY  NOTIONS  FOR  YOUNGER  PUPlLS. 

4.   Show  the  difference  in  meaning  between  the  expressions: 
12a-'7b-^'ic     and     12«-(T^  +  3c); 
(a-\-b)x(c-\-  d)     and     a-\-(hx6)  +  d\ 
and,  if    a  =  4,     Z»  =  3,     c  =  2,     fZ  =  6,     find  their  values. 

The  use  of  single  letters  to  stand  for  words,  and  of  signs  to 
denote  operations  and  relations,  constitute 

A    KIND    OF    SHORT-HAND    AVRITIXG, 

and  the  resulting  brevity  of  expression  is  of  great  advantage 
in  many  ways.     For  example,  coin  pare  these  two  statements  : 

1.  I  want  the  result  of  adding  to  a  number  three  times 
itself,  subtracting  twice  the  number  from  the  sum,  increasing 
the  remainder  by  five  times,  and  by  four  times,  the  same 
number,  and  finally  subtracting  seven  times  the  original 
number  from  the  last  sum. 

2.  «  +  ?>a  —  2a  -\-5a  +  4a  —  7a  ;  wherein  a  stands  for  the  num- 
ber, and  the  additions  and  subtractions  are  indicated  by  the 
signs  +  and  — . 

In  the  last  form  the  result,  4a,  i.e.,  four  times  the  original 
number,  is  seen  at  a  glance. 

So,  let  a\  y  stand  for  any  two  numbers,  of  which  x  is  the 
larger,  and  express  in  algebraic  form  : 

the  sum  of  the  numbers;  their  difference;  their  product; 

the  proper  fraction  got  by  dividing  one  by  the  other;  the 
improper  fraction; 

the  product  of  the  smaller  number  by  their  difference; 

the  sum,  the  difference,  and  the  product  of  their  squares; 

the  squares  of  their  sum,  their  difference,  and  their  product; 

the  i)r()dnct  and  quotient  of  their  sum  by  their  difference; 

the  product  of  the  squares  of  their  sum  and  their  difference; 

th(!  product  of  the  sum  and  ditl'erence  of  their  squares; 

the  sum  and  the  difference  of  their  cubes; 

the  cu})e  of  their  sum  and  of  their  difference. 
So,  tell  in  words  what  these  algebraic  exj)ressi()ns  mean: 
2{x  +  yy,     xy{x  +  y);        ^(^r-y');     x'-hy';  xhr, 

(x^-yy;     ^G-^  +  y  +  O);     10 -rn/;        K2.r  +  3//);     xUf', 


PRIMARY   NOTIONS  FOR  YOUNGER  PUPILS.  Si 

The  examples  above  are  cases  of 

TRAXSLATIOX  IXTO  ALGEBRAIC  FORMS. 

The  pupil  may  also  translate  the  answers  to  these  questions: 

1.  At  a  dollars  a  pair  what  w^ill  5  pairs  of  gloves  cost  ?  l 
pairs?     2^  pairs?     «  pairs  ?     2«  pairs? 

How  many  pairs  can  be  bought  for  c  dollars  ?  for  I  dollars  ? 

2.  An  orchard  contains  r  rows  of  t  trees  each,  and  each  tree 
bears  h  barrels  of  apples:  how  many  barrels  of  apples  are 
raised  ?  and  what  is  their  value  at  d  dollars  a  barrel  ? 

3.  A  man  bought  three  books,  P'^^ji^^g  ^^  dollars  for  the  first 
book,  h  times  as  much  for  the  second  as  for  the  first,  and  for 
the  third  c  times  as  much  as  for  the  other  two:  what  was  the 
cost  of  each  book  ?  of  all  of  them  ? 

4.  A  bill  of  groceries  shows  t  pounds  of  tea  at  x  cents  a 
pound,  .9  pounds  of  sugar  at  y  cents,  and  c  pounds  of  coffee  at 
z  cents:  what  is  the  whole  cost  in  cents?  in  dollars  ? 

5.  At  :/•  dollars  a  3^ard,  what  will  c  yards  of  cloth  cost?  If 
the  same  cloth  be  sold  for  //  dollars  a  yard,  and  something  be 
gained,  what  relation  has  y  io  x*^  How  much  is  gained  on 
one  yard  ?  on  the  c  yards? 

6.  A  square  field  is  /  rods  long:  how  wdde  is  it?  what  is 
its  area  ?  how  long  a  fence  is  needed  to  enclose  it?  how  much 
more  to  divide  it  into  four  equal  square  fields?  how  much  to 
divide  it  into  four  equal  rectangular  fields  ? 

Xote  that  the  numerical  values  are  not  determined,  only 
the  relations  between  the  length,  breadth,  area,  length  of 
fences,  and  so  on,  relations  which  hold  good  whatever  num- 
ber/stands for  ;  and  state  these  relations  in  words. 

7.  A  can  do  a  piece  of  work  in  a  days,  B  in  ^  days,  C  in  c 
days:  what  part  of  the  work  can  A  do  in  one  day  ?  B  ?  C  ? 
A  and  B  working  together  ?   B  and  C  ?   C  and  A  ?   A,  B, 

andC?      What  does  -  stand  for  ?    ]^?    -?    -  +  1^?    i  +  -? 
a  b       c       a      0       0      c 

c      a        a      0     c 


^G-H^-  4-4)'  -&4)' 


xii        PRIMARY  NOTIONS   FOR  YOUNGER  PUPILS. 

i..(i4+L).  ^-{'-,u'-)?  1-3(1+1+1)? 

\a      b      cl  \a      b      c  I  \a      b      c ) 

8.  If  rt  =  lG,  b  =  V2,  c=6,  fiud  the  value  of  each  of  the 
expressions  above;  and  find  what  part  of  the  work  remains 
undone  after  A,  B,  and  C  have  worked  together  three  days. 

9.  If  the  figures  used  in  writing  a  two-figure  number  be 
ciy  b,  the  number  is  written  10a  +  b\  express  half  the  number; 
the  square  root  of  it;  a  number  that  is  c  units  less  than  this 
number;  a  number  that  is  c  units  less  than  a  fourth  part 
of  it ;  the  number  whose  figures  are  5,  a. 

10.  If  a  boy  be  y  years  old  now,  how  old  was  he  a  year  ago  ? 
two  years  ago?  how  old  will  he  be  /"  years  hence?  how  old 
when  his  age  is  doubled  ?  what  is  a  third  of  his  age  ?  two 
thirds  of  what  it  will  be  h  years  hence  ? 

11.  A  rectangular  pile  of  wood  is  a  feet  long,  b  feet  wide, 
and  c  feet  high:  how  many  square  feet  are  there  in  the  top  ? 
in  one  end  ?  in  one  side  ?  how  many  cubic  feet  in  the  pile  ? 
how  many  cords  ?  what  is  its  value  at  d  dollars  a  cord  ? 

12.  If  X  be  the  larger  of  two  numbers  and  d  their  differ- 
ence, what  is  the  smaller  number?  If  x  be  the  smaller  num- 
ber and  d  their  difference,  what  is  the  larger  ? 

13.  If  jj  be  the  product  of  two  numbers  and  n  be  one  of 
them,  what  is  the  other  ?  If  </  be  the  quotient  and  n  one 
number,  what  is  the  other? 

14.  If  a  bushels  of  wheat  cost  a  dollars,  what  is  tlie  price 
of  one  bushel  ?  If  c  bushels  of  this  wheat  sell  for  //  dollars, 
what  is  gained  or  lost  on  one  bushel?  on  the  whole  lot? 

15.  At  ])  cents  a  yard  what  will  it  cost  to  plaster  the  walls 
and  ceiling  of  a  room  a  feet  long,  b  feet  wide,  c  feet  liigh? 

16.  If  a  man  can  row  lu  miles  an  hour  in  still  water,  what 
progress  will  he  make  rowing  with  a  tide  that  runs  /  miles  an 
hour?  rowing  jigainst  the  tide  ? 

17.  A  man  has  1)1  miles  to  walk  in  //  hours;  he  walks  r 
miles  an  hour  for  the  first  b  hours:  how  fast  must  he  walk 
the  rest  of  the  way? 


PRIMARY   NOTIONS  FOR   YOUNGER  PUPILS.       XUl 

18.  A  merchant  began  bnsiiJess  with  a  capital  of  a  dollai-s; 
the  first  year  he  doubled  his  money;  the  second  year  he 
gained  b  times  the  original  capital;  the  third  year  he  lost 
/  dollars  and  died,  leaving  c  children;  the  cost  of  settling  the 
estate  was  s  dollars:  how  much  did  each  child  receive  ?  Is  it 
possible  that  there  was  nothing  to  divide? 

19.  A  and  B  start  from  the  same  place  and  walk  in  the 
same  direction,  A  at  a  miles  an  hour  and  B,  h  miles:  how  far 
apart  are  they  at  the  end  of  an  hour?  at  the  end  of  li  houi-s? 
Do  you  know  from  this  statement  which  man  walks  the  faster? 

So,  if  they  go  in  opposite  directions? 

So,  if  starting  from  two  places  Jc  miles  apart,  they  walk 
towards  each  other?  if  away  from  each  other? 

THE  USE  OF  LETTERS  IN  SOLVING  PROBLEMS. 

Sometimes  statements  are  made  of  such  a  nature  that  the 
actual  value  of  an  unknown  number,  at  first  represented  by  a 
letter,  may  afterwards  be  found. 

For  example,  to  find  a  number  such  that  if  its  double  and 
its  quadruple  be  added  to  it,  the  sum  shall  be  63. 

This  problem  is  easily  solved  by  arithmetic,  as  follows: 

If  to  a  number  its  double  be  added  the  sum  is  three  times 
the  number;  and  this  sum  increased  by  four  times  the  num- 
ber is  seven  times  the  number,  which  is  63.  If  then  63  be 
seven  times  the  number  sought,  that  number  is  a  seventh 
part  of  63,  i.e.,  the  number  is  9. 

The  words  the  iiumhcr  are  used  seven  times  in  this  state- 
ment, and  the  process  may  be  shortened  by  writing  some 
letter,  as  x,  for  these  words  and  expressing  the  operations 
and  relations  by  signs.  In  this  symbolic  language,  2.r  stands 
foi  the  double  of  the  number  and  4a;  for  its  quadruple;  and 
it  is  easy  to  translate  into  algebraic  forms  what  the  statement 
above  gives  in  words: 

a  number!  increased  by  | its  double | and | its  quadruple | gives 1 63 ; 
X  +  2x  +  4:X  =63 

I.e.,     x  +  '2x  +  ^x~^^,     7a:  =  63,     a;  =9,     as  before. 


xiv       riiLMARY   NOTIONS  FOR  YOUNGER   PUPILS. 

So,  if  the  sum  of  the  ages  of  father,  mother,  sod,  and 
daughter  be  100  years,  if  the  boy  be  twice  as  old  as  his  sister, 
the  mother  four  times  as  old  as  her  son,  and  the  father^s  age 
be  three  times  the  sum  of  his  children's  ages;  these  facts  may 
be  expressed  in  algebraic  form  by  writing  x  for  the  girl's  age, 
2x  for  the  bey's,  lour  times  2x,  i.e.,  Sx,  for  the  mother's,  ^x 
for  the  sum  of  the  children's  ages,  and  three  times  3^\  i.e.,  9x, 
for  the  father's  age;  then  9:c  +  8a:4-2a;  +  a:  =  100,  20.?:  =  100, 
x^b;  and  the  girl  is  five  years  old,  the  boy  ten,  the  mother 
forty,  and  the  father  forty-five. 

So,  if  a  man  pay  $45  for  a  saddle  and  a  bridle,  and  the 
saddle  cost  four  times  as  much  as  the  bridle;  then  x  may  stand 
for  the  cost  of  the  biidle,  4.6-  for  that  of  the  saddle,  and  the 
equation  4.i:  +  :i  =  45  expresses  all  the  facts  set  forth  in  the 
statement  of  the  problem.  From  this  equation  comes  DX  =  4ib, 
x  —  d,     4.c=36,     the  values  sought. 

For  further  practice  the  pupil  may  choose  some  letter  or 
other  character  (a  star  or  the  picture  of  a  dragon-fly  would 
serve  just  as  well  if  as  easily  made  and  read)  to  stand  for  one 
of  the  unknown  elements  of  the  problem,  express  the  other 
unknown  elements  as  multiples  or  parts  of  the  letter,  trans- 
late the  word-statements  into  algebraic  forms,  solve  the  result- 
ing equations,  and  finally  determine  the  values  sought. 

1.  A  and  R  together  liave  $100,  and  B  has  three  times  as 
much  as  A;  how  much  has  each  of  them  ? 

2.  A  man  paid  $24  for  a  hat,  a  vest,  and  a  coat;  the  vest 
cost  twice,  and  the  coat  three  times,  as  much  as  the  hat:  what 
was  the  cost  of  each  of  them  ? 

3.  Divide  108  into  three  parts  such  that  the  second  part 
shall  be  twice  the  first,  and  the  third  three  times  the  second. 

4.  Three  ti-ees  together  bear  32  bushels  of  apples;  the  second 
tree  bears  twelve  times  as  much  as  the  fiist,  aiul  the  third  a 
fourth  part  of  the  yield  of  the  second:  how  many  bushels 
does  the  first  tree  bear? 

5.  John  is  three  times  as  old  as  Henry,  and  the  difference  of 
their  ages  is  12  years:  how  old  is  each  of  them? 


PRIMARY  NOTIONS  .FOR   YOUNGER  PUPILS.         XV 

6.  The  difference  of  two  numbers  is  seven  times  the  less,  and 
if  four  times  the  less  be  taken  from  the  greater,  the  remainder 
is  24:  what  are  the  numbers? 

7.  B  is  three  times  as  old,  and  C  four  times  as  old,  as  A,  and 
the  sum  of  their  ages  is  s  yeai  s :  how  old  is  each  of  them  ? 

THE    MATURE   OF    EQUATIONS. 

In  the  problems  that  have  been  solved  above,  some  number, 
or  the  difference  of  two  numbers,  or  the  sum  of  two  or  more 
numbers,  has  always  been  given  as  equal  to  some  other  num- 
ber. Such  expressions  of  equality  are  called  equations,  and 
their  nature  and  uses  should  be  well  understood.  The  two 
parts  of  an  equation,  separated  by  the  sign  of  equality,  are  its 
members,  and  the  numbers  that  make  up  the  two  members 
and  are  joined  by  the  signs  +  and  —  are  its  terms. 

An  equation  of  itself  cannot  be  said  to  have  any  sign  or 
value;  it  is  like  a  balance,  one  member  being  the  thing  weighed 
and  the  other  member  the  weights.  The  sign  of  equality  cor- 
responds to  the  pivot. 

Regarding  an  equation  in  this  light,  it  is  evident  that  the 
same  number  may  be  added  to,  or  subtracted  from,  both 
members  of  an  equation,  and  that  both  members  maybe  mul- 
tiplied, or  divided,  by  the  same  number,  and  the  resulting 
numbers  be  still  equal. 

For  example,  if     2;/;  =  6,     then     a:  =  3     and     6:c  =  18. 

"So,  if    x—Q  —  2x,     then     x-\-2x=Q     and     x  =  2. 

So,  if  a;  +  3  =  24  —  )tx,  then  3  may  be  subtracted  from  both 
members,  and  2.c  be  added,  and  there  results  the  equation 
3x=:21;    then  both  members  may  be  divided  by  3  and    x=l. 

So,  if    \x  =  Zy    then    a;  =12;     and  if    \/x  =  3,    then    x=:9. 

In  the  same  manner  the  pupil  may  solve  the  equations  below: 
1.  4-2:^  =  8-62:.         2.  3x-4:8  =  x  +  12.      3.  I  =  10-f2:. 
4.  ^(a;-6)  =  20.  5.  5  +  C>.t  =  22:  +  17.      G.  ^.r-3  =  9. 

7.  3a:=-/9.  8.  5(z-6)=^(17-a:).     9.  |/.«  +  5  =  2 |/.r  +  3, 

10.  \/Zx-2  =  l.     11.   ^x-\^pTi,  12.  v'a;  +  l=yi6. 


XVI       PRIMARY  NOTIONS  FOR  YOUNGER  PUPILS. 

THE    USE    OF    LETTERS    AND     EQUATIONS    IN    STATING    RULES 
AND    PRINCIPLES. 

The  familiar  rules  and  princij^les  of  arithmetic  may  often 
be  stated  in  algebxaic  language.  One  advantage  of  such 
statements  is  their  brevity  and  clearness,  as  noted  above,  and 
the  consequent  ability  of  the  pupil  to  see  the  whole  at  once, 
and  to  note  the  relations  of  all  the  parts. 

For  example,  that  the  value  of  a  fraction  is  not  changed  if 
hoth  its  terms  he  7)iultiplied  hy  the  same  number,  a  statement 
of  two  lines,  is  exp^-essed  in  algebraic  form  by  the  use  of  six 

letters  and  three  sVns :  it  is  written     7-=^^ ,     wherein  — 

°  b      bxm  b 

stands  for  any  fraction  whatever  and  m  for  any  multiplier. 

So,  the  equation      —^-r- — -     means  that  hoth  terms  of  a 

fraction  maij  he  divided  by  the  same  number  ivithout  changing 
its  value. 

So,  the  proposition — if  four  numbers  be  in  proportion,  the 
product  of  the  extremes  is  equal  to  the  product  of  the  means,  is 
expressed  by  writing     if    a  :b  =  c  :d,     then     axd=bxc, 
wherein  a,  b,  c,  d,  are  any  four  numbers. 

So,  that  the  retnaiiider  is  not  changed  if  both  minuend  and 
subtrahend  be  increased  by  the  same  number  is  expressed  by 
the  equation  {m  +  n)  —  (s  +  n)  =  m  —  s  =  r,  wherein  m  is  the 
minuend,  s  the  subtrahend,  r  the  remainder,  n  any  number. 

a  c 

So,     —  and  -  may  stand  for  any  two  fractions,  and  the  rule 

for  their  addition  may  be  briefly  written     ^^ -7 — -. -. 

The  pupil  may  in  like  manner  write  the  rules  for  subtract- 
ing one  fraction  from  another;  for  multiplying  two  fractions 
together;  for  dividing  one  fraction  by  another. 

So,  he  may  express  the  relations  between  the  dividend, 
divisor,  quotient,  and  remainder,  in  division. 

So,  lie  may  write  down  the  w^ell-kuown  properties  of  pro- 
portion which  are  (expressed  by  tiie  words  altcrnidion,  inver^ 
sion,  con)2)osition  and  division. 


ALGEBRA 


I.   THE  PRIMARY  OPERATIONS  OF  ARITHMETIC. 


The  pupil  who  begins  the  study  of  algebra  is  already  fam- 
iliar with  the  simpler  operations  of  arithmetic,  and  has  given 
some  thought  to  the  reasons  for  those  operations.  As  algebra 
is  but  a  larger  arithmetic,  he  may  well  stop  to  review  the  funda- 
mental principles,  and  try  to  fix  in  his  mind  more  precise 
notions  of  what  is  to  be  the  basis  of  his  larger  knowledge. 

As  distinguished  from  geometry,  algebra  treats  of  numbers 
and  of  numbers  only;  and  this  chapter,  after  the  definition  of 
number  and  the  distinction  between  the  different  kinds  of 
simple  numbers,  goes  on  to  define  and  discuss  the  operations 
of  multiplication  and  division,  of  addition  and  subtraction, 
and  of  involution  and  evolution. 

Algebra  differs  from  arithmetic  chiefly  in  this  :  that  while 
in  arithmetic  every  number  has  a  definite  and  fixed  value,  and 
the  numerical  expression  of  value  is  the  thing  always  sought, 
in  algebra  the  expression  of  the  relation  of  numbers,  or  of 
some  general  principle,  is  often  what  is  aimed  at,  the  nu- 
merical computation  being  an  arithmetic  operation  that  may 
be  performed  later  or  omitted. 

Algebra  differs  still  further  from  arithmetic  in  the  use  of  a 
set  of  symbols  that  constitute  a  language  of  its  own,  and 
among  the  advantages  of  this  symbolic  language  are  these: 
clearness,  brevity,  and  generality  of  statement;  the  ability  to 
mass  directly  under  the  eye,  and  thus  to  bring  before  the 
mind  as  a  whole,  all  the  steps  in  a  long  and  intricate  investi- 
gation; and  the  facility  of  tracing  a  number  through  all  the 
changes  it  may  undergo. 


.2;        ;itti^  PRIM j\PY   OPERATIONS   OF   ARITHMETIC.  [I. 

§1.    NUMBER. 

The  subject-matter  of  Algebra  is  luimber,  and  numbers 
come  from  counting  and  measuring  ;  they  answer  the  questions 
lioio  many  and  how  much.  Anything  that  is  to  be  measured 
is  a  quantity,  and  the  result  of  the  measurement  is  a  number. 
E.g.,  if  a  boy  count  the  apples  in  a  basket  the  answer  is  a 

number,  say  twelve. 
So,  he  may  measure  the  side  of  a  room  with  a  yard  stick  and 
find  how  many  times,  say  twelve,  a  single  yard  is  con- 
tained in  the  whole  length. 

INTEGERS   AND   FRACTIONS. 

if  the  things  counted  be  entire  units,  or  if  the  thing  meas- 
ured contain  the  measuring  unit  exactly  so  that  there  is  no 
remainder,  tlie  number  is  an  integer;  but  if  tlie  measurement 
can  be  completed  only  by  using  some  part  of  the  first  unit  as 
a  new  unit  of  measurement,  the  resulting  number,  for  the 
portion  so  measured,  is  a  simiile  fraction. 
E.g.,  if  a  side  of  a  room  contain  a  yard  twelve  times  with  a 
remainder  that  contains  a  third  part  of  a  yard  twice, 
the  whole  length  is  twelve  yards  and  two  thirds. 

CONCRETE   AND   ABSTRACT   NUMBERS. 

In  the  examples  above,  the  number  is  closely  associated  with 
the  things  counted  or  measured,  and  the  whole  answer  to  the 
question  hoio  many  is  twelve  apples,  or  twelve  and  two-thirds 
yards.  Such  numbers  are  concrete  numbers,  and  concrete 
numbers  may  be  defined  as  measured  quantities. 
E.g.,  twelve  apples,  five  days,  twenty  pounds. 

But  an  abstract  number  implies  some  operation,  and  it  may 
be  called  an  operator.  The  simplest  operations  are  those  of 
repetition,  such  as  doubling,  tripling,  and  quadrupling;  and  of 
partition,  such  as  halving  and  quartering. 
E.g.,  if  a  child  pick  up  twelve  blocks,  lie  may  pick  np  one 
block,  tlien  another,  and  another  till  he  has  twelve  of 
them, 


§1)  KUMBEB.  3 

and  the  operation  that  the  operator  twelve  calls  for  is  repe- 
tition of  the  single  act  of  picking  up  a  block. 

So,  he  may  part  his  blocks  into  two  equal  piles,  or  cut  an  apple 
into  two  equal  parts;  the  operator  is  two  and  tlie  opera- 
tion  is,  j^artitiuii. 
That  thing  which  is  acted  upon  is  the  u7iU, 

E.g.,  the  single  block  in  the  first  case,  and  the  whole  group  of 
blocks,  or  the  apple  in  the  other,  is  the  unit. 
The  combination  of  a  unit  and  a  numerical  operator  forms 

a  concrete  number. 

QUESTIONS. 

1.  Give  some  number  that  answers  the  question  hoiv  long  a 
time,  the  question  Jioiu  niucJi,  the  question  hoiv  far. 

2.  Xame  the  different  standard  units  of  time,  and  state 
their  relation  to  each  other. 

So.  of  length,  of  area,  of  volume,  and  of  weight. 
Of  these  units,  which  are  natural?  which  artificial? 

3.  Can  the  question  Iiotv  ill  be  answered  by  giving  a  number? 
So,  of  the  questions  Jiow  difficult,  how  good,  hotu  beautiful? 
So,  of  the  questions  hoio  warm,  lioiu  bright,  hoio  strong? 

4.  By  what  unit  can  a  room  that  is  twelve  and  two-thirds 
yards  long  be  measured,  so  as  to  give  an  integer  result  ? 

5.  If  the  same  thing  be  measured,  what  is  the  effect  on  the 
number  if  a  smaller  unit  be  used  ?  if  a  larger  unit? 

6.  Count  two  dozen  eggs  with  six  eggs  as  the  unit;  what  is 
the  operator?  with  one  egg  as  the  unit,  what  is  the  operator  ? 

7.  Express  the  distance  six  feet  with  three  inches  as  the 
unit,  then  take  a  smaller  unit  and  a  smaller;  how  far  can  this 
process  of  reducing  the  unit  be  carried  ? 

So,  if  the  unit  be  taken  larger  and  larger  ? 

8.  Draw  a  line  six  inches  long,  and  express  three  quarters 
of  it  with  half  an  inch  as  the  unit.  How  many  distinct  units 
are  here  used  ?    How  many  operations?  how  many  operators? 

Which  of  these  operations  are  repetitions  ?  w^hich  partitions? 


4  THE  PRIMARY   OPERATIONS   OF   ARITHMETIC.  II, 

EQUAL   NUMBERS. 

That  two  concrete  numbers  be  equal,  it  is  necessary  that  the 
things  counted  or  measured  be  of  tlie  same  kind,  and  that,  if 
the  units  be  taken  of  the  same  magnitude,  there  shall  be  as 
many  units  in  tlie  one  group  as  in  tlie  other. 

If  there  be  more  units  in  one  group  than  in  the  other,  tlie 
first  number  is  larger  than  the  other ;  if  fewer,  it  is  smaller. 

That  two  abstract  numbers  be  equal,  it  is  sufficient  that 
when  operating  on  the  same  unit  they  shall  give  the  same 
result;  examples  of  such  numbers  appear  later. 

That  two  numbers  are  equal  is  shown  by  the  sign  =  ;  that 
they  are  unequal  by  =^  ;  that  the  first  is  larger  than  the  other 
by  the  sign  >  ;  that  it  is  smaller  by  <  . 

EXPRESSION    OF    NUMBERS. 

In  algebra  numbers  are  represented  by  Arabic  numerals, 
as  in  arithmetic;  they  are  also  often  represented  by  letters. 
E.g.,  in  questions  about  interest,  p  may  stand  for  principal,  r 
for  rate,  t  for  time,  i  for  interest,  a  for  amount. 

If  tliere  be  four  promissory  notes,  the  four  principals  may 
be  writen  p',  p" ,  p'",  p'",  read  p  jjrime,  p  second,  p  third, 
p  fourth,  or  pi ,  p^ ,  p^ ,  p^ ,  read  p  one,  p  two,  p  three,  p  four  ; 
and  the  corresponding  rates  and  times  would  then  be  written 
r',  V,  r'\  f\  r"\  t"\  r'\  t'\  or  r,,  t,,  r„t,,  r„  t„  r„  t,. 

Letters  or  figures  attached  to  other  letters  are  indices. 

It  is  customary  to  express  a  concrete  number  by  writing 
the  operator  first,  and  after  it  the  unit. 
E.g.,  twelve  blocks,  half  an  apple,  two  thirds  of  a  yard. 

But  the  purposes  of  algebra  are  sometimes  better  served  by 
writing  the  unit  first  and  following  it  with  the  sign  of  opera- 
tion and  the  operator. 

E.g.,  block  X  12,  wherein  the  cross  means  repetition. 
So,  apple  :  2,  wherein  the  colon  means  ])arlitioii,  or  apple  x  1/2. 
So,  yard  :  3  x  2,  or  yard  x  2/3,  means  that  the  yard  is  divided 
into  three  equal  parts,  and  two  of  them  are  taken. 


§  1  ]  NUMBER. 


QUESTIONS. 

1.  Find  other  concrete  numbers  that  are  equal  to  three 
hundred  minutes;  to  thirty-six  inches;  to  half  a  mile. 

2.  If  there  be  two  farms  each  of  one  hundred  acres,  are 
they  equal  in  area  ?  have  they  the  same  sha23e  ?  are  they 
equal  in  value  ? 

3.  If  there  be  two  farms,  one  of  fifty  acres  worth  a  hundred 
dollars  an  acre,  and  the  other  of  a  hundred  acres  worth  fifty 
dollars  an  acre,  in  what  respect  are  the  two  farms  equal? 

What  two  concrete  numbers  are  now  equal  ? 

What  is  the  unit,  and  what  are  the  operators  that  give  these 
two  equal  concrete  numbers?  are  they  found  by  precisely 
the  same,  or  by  different,  operations? 

4.  One  room  is  four  yards  by  six  and  another  three  yards  by 
eight ;  will  the  same  carpet  cover  both  floors?  can  it  be  made 
up  in  two  parts  so  as  to  cover  either  floor  at  will  ? 

In  what  respect  are  the  two  floors  equal?  in  what  unequal? 

5.  If  the  answer  to  the  question  lioiv  long  be  six  days,  which 
is  thought  of  first,  the  unit  day  or  the  number  six  ? 

6.  If  the  length  of  a  room  be  sought,  which  is  found  first, 
the  yard  stick  or  the  length  in  yards  ? 

7.  What  does  a  salesman  do  when  he  measures  off  50  yards 
of  carpet  ?  so,  when  he  cuts  this  carpet  into  six  equal  breadths  ? 

In  each  of  these  operations  what  is  the  unit  and  what  is  the 
operator  ? 

8.  Following  the  same  system  of  notation  as  in  interest^ 
how  can  the  area  of  a  rectangular  house-lot  be  expressed  if  the 
length  and  breadth  be  known  in  feet  ?  How  can  the  cost  be 
expressed  if  the  price  per  square  foot  be  also  given? 

So,  the  cost  of  a  block  of  marble  whose  length,  breadth, 
and  height  are  known,  and  the  price  per  cubic  foot  ? 

9.  Write  an  expression  to  show  the  order  of  procedure  if  a 
grocer  sell  ten  eggs  to  each  of  two  customers  for  three  succes- 
sive days;  ten  eggs  to  each  of  three  customers  for  two  days. 


6  THE  PRIMARY  OPERATIONS  OF  ARITHMETIC.  [I. 

§2.   MULTIPLICATION  AND   DIVISION. 

The  product  of  a  concrete  number,  the  ^mdtiplicand,  by  an 
abstract  number,  the  multiplier ,  is  the  result  of  the  repetition 
or  partition  of  the  concrete  number  by  the  other  used  as  an 
operator.  The  product  is  a  concrete  number  of  the  same  kind 
as  the  multiplicand.  Multiplication  thus  embraces  halving 
and  quartering  as  well  as  doubling,  tripling,  and  quadrupling. 
E.g.,  $5  X  10  =  $50,     $50  :  10  =  $5,     or  $50  x  1/10  =  $5. 

The  product  of  two  concrete  numbers  is  an  absurdity. 
E.g.,  the  product  $2x5  days  is  impossible; 
but  if  a  man  earn  $2  a  day,  in  5  days  he  earns  $10, 
i.e.,  $2  X  5  =  $10,  wherein  5,  not  5  days,  is  the  m^ultiplier. 

The  solution  rests  on  the  well-recognized  relation  between 
the  time  and  the  wages  earned :  as  five  days  is  five-fold  one 
day,  so  the  wages  of  five  days  is  five-fold  the  wages  of  one  day. 

The  product  of  two  or  more  abstract  numbers  is  an  abstract 
number  that  gives  the  same  result  when  operating  on  a  unit 
as  is  obtained  when  the  unit  is  multiplied  by  the  first  of  the 
given  numbers,  the  product  so  found  multiplied  by  the  second 
number,  and  so  on  till  all  the  numbers  are  used.  The  num- 
bers SiYG  factors  of  the  product. 
E.g.,  if  of  three  men  A,  B,  C,  A  has  $50,  B  twice  as  much  as 

A,  and  0  three  times  as  much  as  B; 
then  B  has  $50  x  2,  or  $100 ;  and  C  has  $100  x  3,  or  $300, 
i.e.,  $50x2x3  =  $300; 
but  since  $50x6  =  $300, 

therefore  the  product  of  the  two  abstract  numbers  2,  3,  is  the 
abstract  number  6. 

That  the  product  of  two  or  more  factors  is  to  be  used  in- 
stead of  the  factors  in  succession,  may  be  expressed  by  enclos- 
ing them  in  brackets  or  placing  them  under  a  bar,  and  these 
signs  indicate  that  the  expt-ession  so  enclosed  is  to  be  first 
simplified  and  tlien  used  as  a  single  number. 
E.g.,  $50  X  (2  X  3)     or     $50  x  2^3. 

The  factors  so  used  form  a  f/roup  of  factors. 


§2]  MULTIPLICATION   AND  DIVISION  / 

QUESTIONS. 

1.  Wlien  is  multiplication  a  process  of  repetition,  and  when 
of  partition  ? 

2.  By  what  process  other  than  mnltiplication  can  the  prod- 
uct of  five  days  by  seven  be  found? 

3.  Can  an  abstract  number  be  multiplied  by  a  concrete 
number  ?  Can  an  abstract  number  be  made  concrete  by  mul- 
tiplication ? 

4.  If  a  concrete  number  be  multiplied  by  an  abstract  num- 
ber, of  what  kind  is  the  product  ?  Can  a  concrete  product  be 
found  without  using  a  concrete  multiplicand  ? 

5.  In  finding  the  area  of  a  rectangle  we  seem  to  multiply 
one  concrete  number,  the  length,  by  another  concrete  number, 
the  breadth,  state  what  is  really  done. 

6.  What  is  the  test  of  equality  for  abstract  numbers? 

7.  How  are  we  convinced  that  the  product  of  the  abstract 
numbers  3  and  2  is  the  abstract  number  6  ? 

Does  this  reasoning  apply  to  this  particular  pair  of  numbers 
only,  or  is  it  in  the  nature  of  a  general  proof  that  applies 
alike  to  every  pair  of  abstract  numbers?  e.g.,  does  it  prove 
that  the  product  of  the  two  abstract  numbers  3  and  5  is  the 
abstract  number  15  ? 

8.  What  is  the  cost  of  10  cases  of  eggs,  each  containing  6 
boxes  that  hold  two  dozen,  at  1^  cents  apiece  ?  Exhibit  the 
factors  in  the  order  named. 

Exhibit  them  in  order  if  the  number  of  dozen  be  first  found 
and  the  price  per  dozen. 

So,  if  the  cost  of  one  case  be  first  found. 

9.  What  change  is  made  in  the  product  by  multiplying  the 
multiplicand  by  some  number  and  then  using  the  same  m.ul- 
tiplier  as  before  ? 

So,  by  multiplying  the  multiplier  and  leaving  the  multipli- 
cand unchanged? 

10.  Name  the  integer  factors  of  12,  of  35,  of  315, 


8  THE  PRIMARY  OPERATIONS  OF  ARITHMETIC.     [I.Th 

MULTIPLICATION"   ASSOCIATIVE. 

The  sign  •.*  means  since  or  because;  /. ,  therefore;  •  •  • ,  and 
so  on;  and  the  letters  q.e.d.,  icliicli  ivas  to  he  proved. 

Theor.  1.  The  product  of  three  or  more  abstract  jiumhers  is 
the  same  number,  however  the  factors  be  grouped. 
E.g.,  let  5,  3,  1/3,  7,  be  four  abstract  numbers; 
then  are  the  products  (5  x  3)  x  (1/2  x  7),  5  x  (3  x  1/2  x  7), 

(5  X  3  X  1/2)  X  7  each  equal  to  5  x  3  x  1/2  x  7. 
For,  to  multiply  a  unit  by  the  product  5x3  gives  the  same 

result  as  to  multiply  the  unit  first  by  5  and  that  product 

by  3;  [df.  prod.  abs.  nos. 

So,  to  multiply  this  product  by  the  product  1/2  x  7  gives  the 

same  result  as  to  multiply  it  first  by  1/2  and   that 

product  by  7; 
and  the  result  of  the  four  multiplications  is  the  product 

unit  X  5  X  3  X  1/2  x  7. 
So,  the  product  unit  x  (5  x  3  x  1/2  x  7)  =  unit  x  5  x  3  x  1/2  x  7. 
and  •.•  the  abstract  products  (5  x  3)  x  (1/2  x  7),  5  x  3  x  1/2  x  7 
do  the  same  work  when  operating  on  a  unit, 
/.  these  abstract  products  are  equal; 
and  so  for  the  other  abstract  products.  q.e.d. 

So,  to  make  tlie  reasoning  general,  let  a,  b,  1/c,  •  •  •  1/k,  I  be 

any  abstract  numbers,  operators  that  mean  repetitions 

and  partitions; 

then  are  the  products  ax(bx  l/c)  x  •  •  •  {l/k  x  /), 

(axb)x  (l/c  X  . .  .  \/k  X  /),  and  all  others  that  may  be 
formed  by  different  grouping  of  the  factors,  each  equal 
to  the  product  ax  b  x  l/c  x  •  •  •  l/k  x  I. 

For,  to  multiply  tlie  concrete  pnxluct  unitxrir  by  the  ab- 
stract product  bxl/c  is  to  multiply  the  concrete 
product  unitxrt  by  the  abstract  number  />,  and  the 
consequent  concrete  product  unit  xaxb  by  tlie  ab- 
stract number  l/c  ;  [df.  prod.  abs.  nos. 

i.e.,  unit  xax(bx  l/c)  =  unit  xaxbx  l/c. 


t,§2]  MULTIPLICATIOX   AND   DIVISION.  9 

So.  for  the  product  of  this  jDroduct  by  the  abstract  products 

that  follow  in  order, 
i.e.,  unit  xax(bx  1/c)  •  •  •  (1/k  x  I), 
=  unit  X  axbx  1/c  •  •  •  1/k  x  I 
=  unit  x(axbx  1/c  •  •  •  1/k  x  I), 
So,  for  the  other  concrete  products. 

.*.  the  abstract  product  of  these  factors  is  the  same,  how- 
ever they  be  grouped.  q.e.d. 
The  principle  proved  in  theor.  1  is  the  associative  pj^incij^U 
of  multiplication  ;  the  theorem  is  sometimes  written,  imiUi- 
plication  is  an  associative  operation. 

QUESTIONS. 

1.  By  actual  multiplication,  show  the  equality  of  the  prod- 
ucts   5  X  1/10  X  2  X  7  X  1/14  X  2,    5x1/10x2x7x1/14x2. 

2.  What  is  a  theorem  ?  what  a  proof  ?  [consult  a  dictionary. 

3.  In  the  proof  of  theor.  1,  for  what  purpose  are  the 
factors  represented  by  letters  ?  Which  of  the  factors  a,  h, 
1/c,  '  •  •  1/k,  I,  indicate  repetitions  and  which  partitions  ? 

4.  Is  the  caption  of  theor.  1  a  statement  of  whose  truth  we 
are  certain  at  first,  or  one  that  must  be  proved  ? 

5.  The  second  paragraph  of  the  proof  ^'then  are  the  prod- 
ucts •  •  •  ^Ms  a  restatement  of  the  last  line  of  the  theorem; 
why  this  restatement?  Are  the  statements  that  follow  known 
to  be  true,  or  must  they  be  proved  ? 

6.  May  the  proof  of  a  theorem  rest  on  statements  that 
"seem  reasonable,"  or  must  it  rest  on  the  authority  of  defini- 
tions and  axioms,  and  of  other  theorems  that  have  been  already 
fully  established  by  means  of  definitions  and  axioms? 

7.  Is  the  theorem  true,  and  the  proof  conclusive,  if  there 
be  but  five  factors?  what  is  the  meaning  of  the  dots? 

8.  State  the  associative  principle  of  multiplication. 

9.  Show  that  the  statement  of  the  theorem  and  its  proof 
depend  directly  and  wholly  upon  the  definition  of  the  product 
of  abstract  numbers. 


10  THE  PRIMARY   OPERATIONS   OF   ARITHMETIC.     H.Th 

^01  MULTIPLICATION  COMMUTATIVE. 

'^tEOR.  2.   Tlie  product  of  tivo  or  more  abstract  numbers  is 
th  lame  ninnber,  in  ivliatever  order  the  factors  he  multiiMed, 

{a)  Ti  )  factors,  m,  n,  both  expressing  repetition; 
then  the  two  abstract  products  m  X7i,  nxm  are  equal. 
For  let  ****...* 

5f€  9|C  ^  %r*««*^ 


3j»  5J»  5jC  %r>***^ 

be  a  collection  of  like  units,  say  stars,  arranged  in  rectangular 
form,  m  units  broad  and  n  units  deep,  so  as  to  form  m 
vertical  columns  and  n  horizontal  rows  ; 
then  *.*  the  concrete  product    star  x  7n   is  the  m  stars  in  one 
row, 
.*.  the  concrete  product    starxmx?i   is  the  stars  in  the 
n  rows, 
i.e.,       all  the  stars  in  the  whole  collection  ; 
and  *.*  the  concrete  product   star  x  u    is  the  w  stars  in  one 
column, 
.*.  the  concrete  product  star  xnx  m   is  the  stars  in  the  m 
columns, 
i.e.,       all  the  stars  in  the  whole  collection. 

.*.  the  abstract  product  m  x  n  does  the  same  work  as  an 

operator  on  a  unit  as  the  abstract  product  n  x  m  ; 
,*.  these  two  abstract  products,    7n  x  n,    n  x  m,   are  equal. 

Q.E.D.         [df.  eq.  abs.  nos. 
(b)   Two  factors,  m,  \/n,  expressing  a  repetition,  and  a 
partition  ; 

tiien  the  two  abstract  products   ?// x  1/?/,    l/nxnr   are  equal. 

For,  let  the  unit  be  divided  into  n  equal  parts,  and  let  each 

part  be  i-epresonted  by  one  of  the  n  stars  in  any  column 

taken  from  the  block  of  stars  shown  above; 

then*.*  in  this  block  of  stars  the  conci-ete  product  (one  column 

X7n)  is  the  stars  in  the  m  columns. 


2.  §2]  MULTIPLICATION    AND   DIVISION.  11 

i.e.,  all  the  stars  in  the  whole  collection,  consisting  of  7i  ^    v7s, 
.*.  the  concrete  product  one  column  x  in  x  l/)i   is  th«   4ars 
in  one  row  ; 
and  '.'the  concrete  product    one  column  x  1/n    isr-ne  star, 
/.  the  concrete  product   one  column  x  I/71  x  ni-  is  the  stars 

in  one  row. 
,*. the  two  abstract  products      mxl/?i     and      l/7ixm, 
when  operating  on  the  same  unit,  do  the  same  work, 
and  are  equal.  Q.e.d. 

QUESTIONS. 

1.  In  the  block  of  stars  used  in  theor.  2,  how  many  of  the 
771  stars  in  a  row  are  not  shown  ?  of  the  }i  stars  in  a  column  ? 

2.  In  the  product  star  x  m  what  is  the  unit  ?  what  is  the 
operator?  why  is  this  product  concrete? 

3.  What  does  the  product  star  x  7]i  x  n  represent  ?  what 
the  product    star  x  ;^  x  7u  ? 

How"  do  you  know  that  these  two  products  are  equal  ? 
Granting  their  equality,  how  does  it  follow  that  the  two  ab- 
stract products     711  X  «,     71X7/1     are  equal  ? 

4.  In  case  (a)  it  is  shown  that  the  product  of  two  abstract 
integers  is  the  same,  in  whatever  order  the  factors  be  taken; 
what  relation  does  this  truth  bear  to  the  complete  theorem  ? 

5.  What  is  the  effect  of  the  operator  ni  acting  on  a  column 
of  stars  as  a  unit  ?  of  the  operator  l/n  acting  on  this  product  ? 

What  is  the  final  result  of  the  two  operations? 

6.  What  is  the  effect  of  the  operator  l/n  acting  on  a  column 
of  stars  as  a  unit?  of  the  operator  /n  acting  on  this  product? 

What  is  the  final  result  of  the  two  operations? 

7.  What  single  fact  proves  that  the  two  abstract  products 
7n  X  l/n,     l/n  x  in     are  equal  ? 

8.  Make  a  formal  statement  of  the  truth  learned  in  case  {h) 
as  if  it  were  a  theorem  by  itself. 

9.  Show  by  a  diagram  that  one  seventh  of  six  equal  lines 
is  equal  to  six  sevenths  of  one  of  these  lines. 


12  THE   PRIMARY  OPERATIONS   OF  ARITHMETIC.     [l,Ta 

(c)  T2C0  factors,  1/m,  \/n,  loth  expressing  partition  ; 
then  the  two  abstract  products  1/my.l/n,    l/nx\/m   are 

equal. 
For,  let  the  unit  be  divided  iuto  m  x  71  equal  parts,  let  each 
part  be  represented  by  a  star,  and  let  the  whole  be  ar- 
ranged in  a  block  of  m  columns  and  n  rows; 

then  •/  in  this  block  of  stars,  tlie  concrete  product 

block  X  1/m    is  the  stars  in  one  column  of  n  stars, 
.*.  the  concrete  product   block  x  1/m  x  \/n   is  one  star; 
and  •.•  the  concrete  product   block  x  1/n    is  the  stars  in  one 
row  of  m  stars, 
.%  the  concrete  product    block  x  1/n  x  ^/m    is  one  star, 
/.the   two   abstract   products,     1/mxl/n,     1/nxl/m, 
when  operating  on  the  same  unit,  do  the  same  work, 
and  are  equal.  q.e.d. 

(d)  Three  or  more  factors,  a,  1/h,  c,-  -  -  1/Tc,  I,   expressing 
repetitions  and  partitions  ; 

then,  however  these  factors  are  arranged  at  first  in  any 
product  of  them,  that  product  is  equal  to  tlieir  product 
when  arranged  in  the  order  a,  1/h,  c,  •  •  •  1/k,  I. 

For  in  any  one  of  these  products,  if  a  be  not  first  it  may  be 
grouped  with  the  factor  before  it,  may  change  places 
with  that  factor  without  changing  the  whole  product, 
and  so  come  to  be  the  first  factor.      [ths.  1,  2  (a,  b,  c). 

So,  1/b  may  change  places  in  turn  with  all  the  factors  that 
stand  before  it  except  a,  and  so  come  to  be  the  second 
factor,  and  so  on  ; 

i.e.,  without  changing  the  proiluct,  the  factors  may  change 
places  successively,  and  como  to  take  the  order  a,  1/b, 

C,'-'\/l',h 

,\  the  abstract  })roduct  (f  x  \/b  x  ^  x  •  •  •  1/kxl,  and 
others  that  may  be  rormcd  by  different  arrangements 
of  the  same  factors,  when  acting  upon  any  unit,  do 
the  same  work,  and  are  equal.  q.e.d. 


^',§2]  MULTIPLICATION    AND   DIVISION.  13 

The  principle  proved  in  tlieor.  2  is  tlie  commutative  princi- 
pie  of  multiplication,  and  the  theorem  is  sometimes  written, 
maltipUcation  is  a  commutative  operation, 

QUESTIONS. 

1.  After  the  proof  of  cases  {a,  h)  what  must  still  be  proved 
before  theor.  2  is  fully  established  ? 

2.  In  case  (c)  what  kind  of  factors  are  considered? 

3.  What  is  the  result  when  the  operators  1/m,  \/n  act  in 
this  order  on  the  group  of  stars  ?  when  the  order  is  reversed  ? 

What  is  proved  by  the  two  products'  being  the  same  ? 

4.  As  a  separate  theorem,  state  what  is  proved  in  case  (c)» 

5.  What  relation  does  the  product  block  •  1/m  bear  to  the 
product   block  •  \/n  when  m  is  4  and  7i  is  3  ? 

6.  Show  that  a  third  of  half  a  circle  is  half  a  third  of  it. 

7.  In  what  respect  do  the  factors  treated  in  case  (c)  differ 
from  those  in  case  («)  ?  in  what  are  they  like  them  ? 

8.  How  does  case  {d)  differ  from  all  the  other  cases  ? 

9.  In  the  proof  of  case  (fZ)  which  of  the  statements  made 
about  the  factor  a  rests  on  the  authority  of  theor.  1? 

Which  of  them  on  that  of  theor.  2  ? 

With  what  kind  of  factors  must  a  be  grouped  that  case  {h) 
may  apply  ? 

10.  ShoAV  that  a  simple  fraction  with  a  numerator  other 
than  unity  could,  by  the  definition  of  a  fraction,  be  regarded 
as  the  product  of  two  factors  of  the  form  a  and  X/h,  and  that 
the  proof  in  case  {fi)  applies  to  such  fractions. 

11.  State  fully  the  truth  proved  in  case  {cX), 

12.  AVhat  is  the  commutative  principle  of  multiplication  ? 

13.  Show  that  the  proof  of  theor.  2  depends  mainly  on  the 
definition  of  equal  abstract  numbers. 

14.  By  multiplication,  show  the  equality  of  the  products 
5  X  1/10  X  2  X  7  X  1/14  X  4,     2  x  4  x  1/10  x  1/14  x  5  x  7. 

Arrange  and  group  these  factors  in  other  ways,  and  show 
that  all  the  products  so  found  are  equal. 


14  THE   PRIMARY  OPERATIONS   OF   ARITHMETIC.   [I.Ths. 

Cor.  1.    The  product  of  tivo  or  more  factors  that  express 

repetitions  and  partitions  is  an  abstract  simple  fractmi. 

For  the  factors  that  express  partitions  may  be  grouped  to- 
gether at  the  left, 

and  their  product,  showing  how  many  equal  parts  the  unit  is 
divided  into,  gives  the  denominator  of  a  fi-action. 

So,  the  factors  that  express  rej)etitions  may  be  grouped  to- 
gether at  the  right, 

and  their  product,  showing  how  many  of  these  equal  parts  are 
taken,  is  the  numerator  of  the  fraction.         q.e.d. 

Cor.  2.  The  product  of  two  or  more  simple  fractions  is  a 
simple  fraction  whose  numerator  is  the  product  of  the  numer- 
ators of  the  factors;  and  whose  denominator  is  the  product  of 
their  denominators. 

Cor.  3.  If  n  he  any  abstract  integer,  then  the  products 
n  X  l/Uy     \/n  X  n     are  each  equcd  to  ^inity. 

Cor.  4.  If  the  numerator  and  denominator  of  a  simple  frac- 
tion he  multiplied  by  the  same  integer,  the  value  of  the  fraction 
is  not  changed  thereby. 

For,  let  n/d  be  a  simple  fraction,  and  multiply  n,  dhy  a; 
then  •.*  a  xl/a=l, 

,;  na/da  =  1/d  x  (1/a  xa)xn  =  n/d.       Q.  e.d.         [cr.  3. 

reciprocals. 
Two  abstract  numbers  whose  product  is  unity  are  reciprocals, 

E.g.,  1/4,  4  ;  3/2,  2/3  ;  3i  2/7  ;  3  x  4,  1/12;  n/d,  d/n. 

Theor.  3.   The  product  of  the  reciprocals  of  two  or  more  ab- 
stract numbers  is  the  reciprocal  of  their  product. 
Let  a,  b,  -  •  '  1/m,  \/n,  •  •  •  p/q,  r/s,  •  •  •  be  any  abstract  num- 
bers ; 
then  is  tlie  product 

\/a  X  \/b  X  '  '  •  m  X  n  X  '  •  •  q/p  x  s/r  •  •  • 
the  reciprocal  of  the  product 

axbx  ' '  '  1/m  x  \/n  x  •  •  •  p/q  x  r/s  •  •  • 


2, 3,  §2]  MULTIPLICATION    AND    DIVISION.  15 

For  the  product  of  these  two  products 

=  (rt  X  1/a)  x{bx  1/b)  X  •  •  •  {l/in  x  vt)  x  {\/n  xn)x  -  -  > 

(p/q  X  q/p)  X  {r/s  x  s/r)  x  •  •  • 
=  lxlx  •.•=!.  Q.E.D.         [ths.  1,  2. 

QUESTIONS. 

1.  What  is  a  corollary  ? 

2.  Why  must  the  product  of  factors  expressing  repetition  or 
partition  be  abstract  ? 

3.  In  what  way  does  cor.  1  rest  on  theor.  2? 

4.  Is  the  product  of  two  or  more  proper  fractions  larger  or 
smaller  than  the  several  factors? 

5.  What  is  the  office  of  the  denominator  of  a  fraction  ? 
What  is  the  derivation  of  the  word  denominator,  and  what  is 
its  primary  meaning?  so,  of  the  word  numerator? 

6.  On  what  does  cor.  2  rest,  directly  ?  indirectly  ? 

7.  By  cor.  2  prove  that  nx\/n  =  \,  then  by  theor.  2  that 
\/nxn  —  \,   and  so  that    n  x\/n  —  \/n  x  n. 

8.  In  proving  cor.  4,  what  use  is  made  of  the  fact  that 
a  X  X/a-l  ? 

9.  What  effect  is  produced  by  multiplying  the  numerator 
of  a  fraction  5/8  by  six  ?  the  denominator  by  six?  both  nu- 
merator and  denominator  by  six  ? 

10.  What  is  the  time  of  a  railway  journey  of  300  miles  if 
the  train  run  10  miles  an  hour?  if  20  miles?  if  30  miles? 

If  the  speed  be  multiplied  by  any  number,  by  what  number 
is  the  time  multiplied  ? 

11.  If  a  certain  piece  of  work  is  to  be  done,  what  is  the  re- 
lation between  the  time  and  the  number  of  men  employed  ? 

12.  What  number  is  its  own  reciprocal  ? 

13.  What  is  the  reciprocal  of  a  proper  fraction  ? 

14.  Without  going  through  the  whole  proof  of  theor.  3, 
state  why  the  product  of  the  reciprocals  of  two  or  more  num- 
bers is  the  reciprocal  of  their  product. 

15.  Can  a  concrete  number  have  a  reciprocal  ? 


16  THE   PRIMARY   OPERATIONS   OF   ARITHMETIC.     [I,Th. 

DIVISION". 

Division  is  an  operation  that  is  the  inverse  of  multiplica- 
tion, i.e.,  when  the  product  of  two  factors,  and  one  of  them, 
are  given,  division  consists  in  finding  the  other  factor.  The 
product  is  now  called  the  dividend,  the  given  factor  is  the 
divisor,  and  the  factor  sought  is  the  quolient. 

If  the  dividend  be  concrete  there  are  two  cases  of  division  i 

1.  The  divisor  abstract  and  the  quotient  concrete. 

2.  The  divisor  concrete  and  the  quotient  abstract. 

The  first  is  a  case  of  partition,  more  or  less  complex,  and  the 
other  a  case  of  finding  how  many  times  one  number  is  con- 
tained in  another  number  of  the  same  kind,  a  ratio. 

E.g.,  •.•  the  product  of  $5  by  4  is  $20, 
.*.  the  quotient  of  $20  by  4  is  $5, 
and        the  quotient  of  $20  by  $5  is  4. 

The  sign  of  division  is  :  or  /,  and  the  order  of  writing  is 
dividend  :  divisor  — quotient,  or  dividend/divisor  =  quotient. 
E.g.,  $20  :  4  =  $5,     $20  :  $5  =-  4,     $20/4  =  $5,     $20/$5  =  4. 

Theor.  4.  If  the  product  of  two  factors  he  multiplied  by  the 
reciprocal  of  one  of  them,  the  result  is  the  other  factor. 
For,  let/,  g  stand  lor  the  two  factors,  and  p  for  their  product; 
i\\Giv'.' p=fxg,  [df.  prod. 

.-.  p  X  l//==/x^  X  l/f=g  x/x  l/f=g.  Q.E.D. 

If  the  divisor  be  abstract,  theor.  4  may  be  written,  the  prod- 
uct of  the  dividend  by  the  reciprocal  of  the  divisor  is  the  quotient. 

Cor.  If  there  be  a  series  of  inultiplications  and  divisions, 
the  final  result  is  the  same,  in  whatever  order  they  be  performed^ 
and  however  the  elements  be  grouped. 

For,  every  division  by  an  abstract  number  is  a  multiplication 

by  the  reci}»rocal  of  the  divisor, 
and  these  multiplications  may  be  perfornuul  in  any  order,  and 

the  factors  be  grouped  in  any  way. 


4,  §2]  MULTIPLICATION   AND   DIVISION.  17 

In  applying  this  corollaiT,  the  pnpil  must  take  care  lest  he 
change  the  office  of  any  element  and  multiply  by  it  where  he 
should  divide,  or  divide  by  it  where  he  should  multiply. 
E.g.,  12  :  3  X  2  X  4  :  8  =  (12  :  3  X  4)  X  (2  :  8) 
:5t  (4  :  3x12)   :   (8x2). 

QUESTIONS. 

1.  Of  the  three  elements  of  a  division,  can  all  be  concrete  ? 
two  concrete  and  one  abstract  ?  two  abstract  and  one  con- 
crete ?  all  abstract  ? 

If  a  man  earn  S4  a  day,  in  what  time  will  he  earn  820? 
Are  not  the  dividend,  divisor,  and  quotient  all  concrete  ? 

2.  In  each  of  the  two  cases  of  division  shown  on  the  oppo- 
site page,  to  what  does  the  divisor  correspond  in  the  multipli- 
cation of  which  the  division  is  the  inverse  ? 

3.  If  the  dividend  be  multiplied  by  some  integer  and  the 
divisor  be  unchanged,  what  is  the  effect  on  the  quotient  ?  if 
the  divisor  alone  be  multiplied?  if  both  be  multiplied  by  the 
same  integer  ?  if  the  dividend  alone  be  divided?  if  the  diviso:' 
alone  be  divided?  if  both  be  divided  by  the  same  integer  ? 

State  these  principles  as  applied  to  the  terms  of  a  fraction. 

4.  Is  a  ratio  a  concrete  or  an  abstract  number? 

5.  On  what  authority  is  it  said,  in  the  proof  of  theor.  4, 
that/x^xl//=^x/xl//?  on  what  that  ^  x/x  1//=^? 

6.  Can  the  second  form  of  stating  theor.  4  be  used  when  the 
divisor  is  concrete?  does  the  first  form  always  apply? 

How  is  the  second  form  applied  in  the  division  of  fractions? 

7.  Why  may  the  multiplications  and  divisions  spoken  of  in 
the  corollary  of  theor.  4,  be  performed  in  any  order? 

Why  may  the  factors  be  grouped  in  any  way  ? 

8.  Replace  the  word  multiplied  by  divided  in  cor.  4  theor.  2 
and  prove  the  resulting  statement.  What  useful  applications 
has   this  corollary  in  reducing  fractious  ? 

9.  Prove  that  the  quotient  of  the  reciprocals  of  two  numbers 
is  the  reciprocal  of  their  quotient. 


18  THE  PRIMARY   OPERATIONS   OF   ARITHMETIC.  1 1, 

§  3.    POSITIVE   AND   NEGATIVE   NUMBERS. 

Sometimes  things  that  are  measured  by  the  same  unit  are 

of  opposite  qualities. 

E.g.,  assets  and  liabilities  are  both  measured  by  the  unit 
dollar. 

So,  dates  a.d.  and  dates  B.C.  are  both  given  in  years. 

So,  the  readings  of  a  thermometer  above  and  below  zero  are 
given  in  degrees. 
In  all  such  cases  if  the  measuring  unit  be  taken  in  the  same 

sense  as  the  thing  measured,  the  resulting  concrete  number 

is  positive;  if  taken  in  the  opposite  sense  the  number  is  nega- 
tive.    In  which  sense  the  unit  shall  be  taken  is  a  matter  of 

custom,  or  of  convenience. 

E.g.,  the  unit  dollai'  may  be  taken  either  as  a  dollar  of  assets  or 
as  a  dollar  of  liabilities : 

if  as  a  dollar  of  assets,  then  assets  are  positive  numbers  and 
liabilities  are  negative  numbers  ; 

if  as  a  dollar  of  liabilities,  then  liabilities  are  positive  and 
assets  are  negative. 

So,  if  distances  measured  towards  the  north  be  positive,  dis- 
tances to  the  south  are  negative, 

i.e.,  if  the  measuring  unit  be  a  northerly  unit,  southerly  dis- 
tances are  expressed  by  negative  numbers. 
Some  things  admit  of  negatives  and  some  do  not. 

E.g.,  time  may  be  counted  backwards  as  well  as  forwards  from 
a  given  date. 

So  may  distances  from  a  given  point. 

So  may  heat  and  cold  from  an  arbitrary  zero. 

So  may  money  of  account,  as  above. 

But  witli  real  dollars,  say  five  of  them,  the  pupil  will  find 
when  he  tries  to  count  past  none — five,  four,  tliree, 
two,  one,  none — tliat  he  is  trying  to  do  wliat  is  im- 
possible. 

So,  a  negative  number  of  persons  by  itself  is  an  absurdity. 


§3]  POSITIVE   AND   NEGATIVE   NUMBERS.  19 

The  primary  notion  of  a  negative  concrete  number  is  that 
of  one  which,  when  taken  with  a  positive  number  of  the  same 
kind,  goes  to  diminish  it,  cancel  it  altogether,  or  reverse  it. 
E.g.,  liabilities  neutralize  so  much  of  assets,  thereby  diminish- 
ing the  net  assets,  or  leaving  a  net  liability. 

QUESTIONS. 

1.  Show  that  the  measuring  unit  of  longitude  maybe  taken 
in  either  of  two  senses,  and  that  whichever  way  the  unit  be 
taken  the  two  kinds  of  longitude  may  be  called  positive  longi- 
tude and  negative  longitude. 

How  are  these  two  kinds  of  longitude  now  distinguished  ? 

2.  If  positive  longitude  be  taken  in  the  direction  of  the 
^un^s  apparent  motion  and  Greenwich  be  the  starting  point, 
is  St.  Petersburg  in  positive  or  negative  longitude  ? 

So,  if  the  direction  of  the  earth's  rotation  be  taken  as  that  ol 
positive  longitude  ? 

3.  If  SI  be  taken  as  the  unit  of  money  possessed,  how  must 
money  spent  be  represented  ?  money  inherited  ?  money  given 
away?  money  owed  ?  money  earned  ?  money  staked  on  a  wager  ? 

4.  If  distances  toward  the  north  be  taken  positive,  how 
must  the  latitude  of  Morocco  be  expressed  ?  of  the  equator  ? 

5.  What  is  the  greatest  possible  positive  latitude  ?  the  larg- 
est negative  latitude? 

6.  What  effect  has  a  negative  concrete  number  when  com- 
bined with  a  larger  positive  number  of  the  same  kind  ?  if  tlie 
negative  number  be  just  as  large  as  the  positive  ?  if  larger? 

7.  If  a  wreck  be  acted  on  by  a  current  setting  northward 
and  by  a  north  wind,  are  the  two  forces  of  the  same  nature  ? 

What  determines  the  direction  in  which  the  wreck  moves  ? 

8.  In  rowing  up  a  river  that  flows  4  miles  an  hour,  what 
progress  is  made  by  a  man  who  can  row  4  miles  an  hour  in 
still  water  ?     What  effect  then  does  the  man's  rowing  produce: 

Show  that  there  are  two  changes,  either  of  which  would 
make  his  progress  visible. 


20  THE  PRIMARY   OPERATIONS  OF  ARITHMETIC.  R 

EXPRESSION   OF   POSITIVE   AND   NEGATIVE   NUMBERS. 

When  denoted  by  Arabic  numerals,  positive  numbers  are 
written  with  the  sign  +  or  with  no  sign,  and  negative  numbers 
with  the  sign  —  ,  before  them.    But  if  a  number  be  denoted  by 
a  letter,  it  is  not  evident  upon  its  face  whether  that  letter 
denotes  a  positive  or  a  negative  number. 
E.g.,  if  the  measuring  unit  be  a  dollar  of  assets,  then +  100,  or 
simply  100  without  the  sign,  means  1100  of  assets,  and 
-100  means  $100  of  liabilities. 
But  N  might  stand  either  for  +100  or  for -100  at  pleasure. 
E.  g.,  if  N  stand  for  + 100,  then  —  N  stands  for  — 100  ; 
and  if  N  stand  for  — 100,  then  — N  stands  for +  100. 

In  this  use  of  the  signs  +  and  — they  are  sigiis  of  quality. 

These  signs  are  also  used  to  indicate  the  operations  of  addi- 
tion and  subtraction,  and  they  are  then  signs  of  operation. 

To  avoid  confusion  in  the  two  uses  of  the  same  signs,  the 
signs  of  quality  may  be  made  small  and  placed  high  up. 
E.g.,  +100  means  1100  of  assets,  and  "100,  $100  of  liabilities. 

But  these  small  signs  are  used  with  the  express  understand- 
ing that  +  is  attached  only  to  numbers  that  are  essentially 
positive,  and  ~  to  those  that  are  essentially  negative.  In  that 
respect  they  may  have  a  meaning  that  differs  from  the  mean- 
ing of  the  large  signs  +  ,  — . 

There  is  a  third  sign,  ±,  made  up  of  the  two,  and  read 
flus  or  minus;  if  written  =f,  it  is  read  minus  or  plus. 
E.g.,  d=3  is  only  an  abbreviated  way  of  writing  the  separate 
expressions  +  3  and  —  3 ;  and  =^  7  is  +  7  or  ~  7. 

AN   ABSTRACT   NEGATIVE   NUMBER  AS   AN   OPERATOR. 

As  an  operator  an  abstract  negative  number  has  two  offices : 

1.  The  repetition  or  partition  of  the  multiplicand. 

2.  The  reversal  of  its  quality; 

and  every  such  number  may  be  regarded  as  itself  the  product 
of  two  factors : 


83]  POSITIVE   AND   NEGATIVE   NUMBERS.  21 

1.  The  absolute  magnitude  of  the  number,  the  te?isor. 

2.  "1,  the  versor. 

E.g.,  -10=+10x-l=-lx+10. 

So,  $1  assets  x  "10  =  810  debts,  and  $1  debts  x  "10  =  $10  assets. 

So,  20  north-miles  X  "10  =  200  south-miles, 

and  20  south-miles  x  "10  =  200  north-miles. 

QUESTIONS. 

1.  If  «=  —  3,  what  is  the  value  of  3«?  of  — 12«? 

2.  Is  it  possible,  in  the  course  of  an  example,  to  have  the 
expression  — 12  men  ?  "  12  men  ? 

Explain  the  difference  between  these  two  expressions. 

3.  What  use  is  made  of  the  signs  +  and  —  in  both  arithme- 
tic and  algebra?  what  use  of  them  is  peculiar  to  algebra? 

4.  If  distances  eastward  from  the  point  where  we  stand  and 
time  after  the  present  moment  be  positive,  and  if  a  passing 
train  be  running  eastward  at  20  miles  an  hour:  show  that 

in  5  hours  it  will  be  100  miles  east  of  us,        +20  x  +5=  +100; 
5  hours  ago  it  was  100  miles  west  of  us,  +20  x  "5=  "100. 

But  if  the  train  be  running  westward,  show  that 
in  5  hours  it  will  be  100  miles  west  of  us,        "20  x  +5=  "100; 
5  hours  ago  it  was  100  miles  east  of  us,  -20  x  "5=  +100. 

5.  Show  that  multiplication  by  a  positive  integer  is  a  case  of 
addition.     What  relation  exists  among  the  numbers  added? 

Then,  with  a  positive  multiplier,  what  relation  has  the  sign 
of  the  product  to  that  of  the  multiplicand  ? 

6.  If  multiplication  by  a  negative  integer  be  regarded  as  a 
case  of  subtraction,  what  are  the  successive  subtrahends  ? 

How  do  their  signs  change  in  the  process?  what  relation  has 
the  sign  of  the  product  to  that  of  the  multiplicand  ? 

7.  What  two  pairs  of  signs  in  the  factors  make  the  product 
positive?  what  two  make  it  negative? 

8.  Show  that  the  product  of  any  even  number  of  negative 
factors  is  positive,  and  that  a  product  can  be  negative  only 
when  it  contains  an  odd  number  of  negative  factors. 


22  THE  PRIMARY  OPERATIONS  OF  ARITHMETIC.  [I, 

§4.    ADDITION  AND   SUBTRACTION. 

If  the  concrete  numbers  added  be  integers  and  simple  frac- 
tions, then,  at  bottom,  addition  is  but  counting,  either  by 
entire  units  or  by  equal  parts  of  a  unit :  on  (forward)  if  posi- 
tive numbers  be  added,  off  (backward)  if  negative  numbers 
be  added.  The  result  is  the  sum,  and  the  sign  is  +,  read  j^lus. 
E.g.,  50  cts.  +  60  cts.  +  90  cts,=$2. 

The  sum  o/two  or  more  abstract  numbers,  operators,  is  an 
operator  that,  acting  on  a  unit  by  repetition,  partition,  or 
reversion,  gives  the  same  concrete  number  for  result  as  if 
the  several  operators  acted  in  turn  upon  the  unit,  and  their 
results,  like  concrete  numbers,  were  then  added. 
E.g.,  7  +  5  =  12; 
for  if  a  man  earn  12  a  day,  then  the  sum  of  his  earnings  for  5 

days  and  for  7  days  is  his  earnings  for  12  days; 
i.e.,  12x5  +  12x7  =  12x12. 
So,  1/5  =  2/15  +  1/15: 

for  if  a  field  be  divided  into  fifteen  equal  house-lots,  a  pur- 
chaser that  takes  two  lots  and  one  lot  has  a  fifth  part 
of  the  whole  field  ;  and  that  whether  the  three  lots  lie 
together  or  apart. 
In  algebra  the  word  addition  is  used  in  a  broader  sense 
than  in  arithmetic,  and  covers  negative  as  well  as  positive 
numbers. 
E.g.,  he  that  has  110,000  assets  and  14000  debts  is  worth  but 

$6000, 
i.e.,       $10,000  assets  +  $4000  debts  =  $G000  net  assets, 
and       + 10,000  +  -  4000  =  +  GOOO. 

Though  the  numbers  to  be  added  must  always  be  of  the 
same  kind,  they  are  often  expressed  by  letters  whose  values 
are  not  known,  or  in  units  whose  values  are  different,  or  which 
cannot  even  be  reduced  to  one  sum:  such  a  group  is  a  poly- 
nomial,  and  the  numbers  to  be  added  are  its  terms. 
E.g.,  5'^  33'"  35"  + 12"  47"*  25^=18'^  21"*;     in  interest  a-p  +  i. 


§4)  ADDITION   AND   SUBTRACTION.  23 

QUESTIOXS. 

1.  By  dividing  up  a  line,  find  the  sum  of  f  and  ^  of  it. 
Hence  find  the  sum  of  the  two  abstract  numbers  f  and  i. 
In  what  common  unit  are  the  two  fractions  expressed  ? 

2.  Before  two  numbers  can  be  added  how  must  they  be  ex- 
pressed ? 

3.  What  is  the  nature  of  an  operator  that  acts  on  a  unit  by 
repetition  ?  by  partition  ?  by  reversion  ?  Xame  some  operator 
that  acts  in  two  of  these  ways;  in  all  three  of  them. 

4.  If  a  unit  be  acted  on  by  two  or  more  operators,  why 
must  the  results  of  the  several  operations  be  concrete  num- 
bers of  the  same  kind  ? 

5.  What  two  arithmetic  operations  may  be  indicated  by  the 
word  addition  in  algebra  ?  If  negative  numbers  be  added  are 
the  minus  signs  signs  of  quality  or  of  operation  ? 

6.  If  240  men  vote  for  a  candidate  and  IGO  vote  against 
him,  what  is  the  sum  of  the  votes  he  receives,  or  his  majoritv? 

Solve,  first  using  signs  of  quality;  then,  stating  the  question 
differently,  solve,  using  signs  of  operation  only. 

7.  When  can  the  addition  of  numbers  be  indicated  but  not 
performed  ? 

What  is  the  sum  of  a,  J,  c,  when  their  values  are  not  known  ? 
when  their  values  are  2,  3,  5  ? 

8.  In  the  expression  4|,  what  sign  of  operation  is  under- 
stood between  the  integer  4  and  the  fraction  f  ? 

So,  between  the  dollars  and  cents  of  818.50. 

So,  in  the  compound  number  12^^  Ib^  35'"20^? 

Show  how  the  operations  so  indicated  may  be  performed. 

9.  Can  the  sum  of  two  numbers  be  smaller  than  one  of 
them  ?  smaller  than  each  of  them  ? 

10.  Draw  a  straight  line  and  mark  a,  b,  two  points  taken  at 
random  upon  it;  then  show  that  the  sum  ab  +  ba  is  naught, 
whichever  direction  be  taken  as  positive. 

So,  take  three  points  A,  b,  c,  in  any  order  upon  a  straight 
line,  and  show  that  ab  +  bc  =  ac  and  that  ab  +  BC  +  ca  =  0. 


2i  THE   PRIMARY  OPERATIONS   OF  ARITHMETIC.   [I,Tb3. 

ADDITION"   COMMUTATIVE   AND   ASSOCIATIVE. 

Theor.  5.  The  sum  of  two  or  more  ahstract  numbers  is  the 
same  immber,  in  whatever  order  the  numbers  be  added  and 
hotuever  they  be  grou'ped. 

For,  let  "^a,  ~b,  c,  »  "  h/k,  =^Z  be  any  abstract  numbers,  posi- 
tive or  negative, 
let  tbese  numbers  act  as  operators  upon  any  unit, 
and  let  the  results  be  grouped  and  added  in  any  way; 
then  '.'the  whole  collection  of  units  and  parts  of  units  is  the 
same,  whichever  unit,  group  of  units,  part,  or  group 
of  parts,  be  counted  first,  whichever  second,  and  so  on, 
,%  the  several  sums  of  these  operators  do  the  same  work 
and  are  all  equal.  [df.  sum  abs.  nos.,  p.  22. 

The  principles  here  established  are  the  commutative  and 
the  associative  2^^'ifioiples  of  addition ;  and  theor.  5  may  be 
written,  additio7i  is  a  commutative  and  associative  operation. 
If  the  pupil  will  cut  card-board  into  any  of  the  common 
geometric  figures,  say  triangles  and  squares,  and  join  them  all 
together  by  placing  them  edge  to  edge  in  various  ways,  he  will 
find  that  the  figures  resulting  from  this  geometric  addition 
are  quite  different  in  form,  and  that,  in  the  sense  of  geometric 
equality,  they  are  not  equal  at  all  ;  but  he  will  find  the  areas, 
the  numerical  results  of  measurement,  to  be  all  equal:  i.e.,  in 
the  geometric  sense  the  figures  are  unequal ;  in  the  algebraic 
sense,  and  for  the  purposes  of  algebra,  they  are  equal. 

Theor.  6.   'Flie  sum  of  two  or  more  simple  fractions  is  a 
simple  fraction,  or  an  integer. 
For,  let  a/b,  c/d,  h/k  •  •  •  be  simple  fractions; 
then  •/  a/b  =  ad/bd=  1/bd  x  ad,    c/d  =  bc/bd  =  1/bd  x  be, 

[th.  2,  cr.  4,  df.  sim.  frac. 

.-.  a/b  +  c/d=l/bd  x  ad+\/bd  x  be  =  \/bd  x  (ad  +  be) 

=  {ad  +  bc)/bd.  [df.  sim.  frac,  df.  ad. 

the  sum  of  this  sum  and  h/k  is  a  simple  fraction  ;  and  so  on. 

E.g.,    2/3  +  3/4  =  17/12;    1/4  +  0/8  =  1. 


5, 6,  §4]  ADDITION   AND   SUBTRACTION.  25 

QUESTIONS. 

1.  What  is  the  associative  principle  of  addition  ? 
What  is  the  commutative  principle  ? 

2.  Do  the  letters  a,}),  c,  -  -  -  stand  for  the  same  numbers  in 
theor.  5  as  in  theor.  1  ?  Can  you  tell  which  of  these  numbers 
are  positive  and  which  negative  ? 

3.  In  the  proof  of  theor.  5,  is  the  value  of  the  entire  collec- 
tion of  units  found  by  adding  the  results  given  by  the  several 
operators,  or  by  the  more  detailed  process  of  counting? 

Which  of  the  operators  are  to  be  applied  by  counting  off? 
Wliich,  by  partition  and  a  later  counting  ? 

4.  What  other  operation  is  commutative  and  associative? 

5.  If  s  stand  for  a  square,  t  for  a  triangle,  c  for  a  half-circle, 
and  r  for  a  rhombus,  is  s^r^-c-\-t  the  same  as  6'  +  ^ +  ?•  +  /? 

If  the  letters  stand  for  the  areas  of  the  figures  are  the  two 
sums  equal  ? 

6.  Cut  a  triangle  from  paper  with  sides  of  different  lengths, 
join  the  mid-points  of  the  sides,  and,  cutting  along  these 
lines,  divide  the  triangle  into  four  triangles  that  may  be  shown 
to  be  all  equal  by  placing  one  upon  another;  combine  these 
triangles  in  all  possible  w\nys:  are  the  geometric  figures  so 
found  equal  ?  are  their  areas  equal  ? 

7.  If  a  merchant  have  various  bills  to  pay  from  a  sum  of 
money  lying  before  him,  show  that  he  has  the  same  amount 
left  after  the  bills  are  all  paid,  whatever  be  the  order  of 
their  payment.     Of  what  principle  is  this  an  example  ? 

8.  If  when  two  fractions  are  added  the  denominator  of  the 
sum  be  a  factor  of  the  numerator,  how  can  the  fraction  be 
more  simply  written  ? 

Is  there  any  number  that  cannot  be  written  in  fraction-form  ? 

9.  In  the  proof  of  theor.  6,  if  the  sum  of  the  operators  ad, 
he  act  on  a  unit,  what  is  the  result  ? 

What  is  the  result  if  ad,  he  act  separately  on  the  unit  and 
the  results  be  then  added? 

What  relations  have  these  two  results  to  each  other  ? 


26  THE  PRIMARY  OPERATIONS  OF  ARITHMETIC.  II,Ths 

MULTIPLICATION   DISTRIBUTIVE   AS   TO   ADDITION. 

Theor.  7.  The  product  of  the  sum  of  two  or  more  abstract 
numlers  by  another  number  is  the  simi  of  the  products  of  the 
first  numbers  by  the  other. 

Let  a,  ~b,  c,  * ' '  h/k,  I,  m  be  any  abstract  numbers; 
then  (a  +  -b-\-c-\-  "  '  +  h/k ■\-l)x m 

=  axm+~bxm-{-cx7n-\'  •  • '  h/k xm-\-lx m. 

For  the  product  (a-^-b-\-c-\ ^-h/k^-l)  xm 

=  7nx(a+~b-[-c+"'  h/k  +  l)  [th.  2,  th.  6. 

=  mxa-{-mx  -b-h  m  xc+  -  -  -  mx  h/k  +  mxl 
=  axm-\-~bxm-\-cxm-\-  •  •  •  h/k  xm-\-lx  m.       [th.  2. 
Cor.   The  prodtict  of  tivo  or  7nore  polynoinials  is  the  sum  of 
the  several  products  of  each  term  of  the  first  factor  by  each  term 
of  the  second  factor  by  each  term  of  the  third  factor,  and  so  on. 
For  the  product  of  two  factors  is  the  sum  of  the  partial  prod- 
ucts of  each  term  of  one  factor  by  each  term  of  the  other, 
and  the  product  of  this  product  by  a  third  factor  is  the  sum 
of  the  partial  products  of  each  term  of  this  product  by 
each  term  of  the  third  factor,  and  so  on.        q.e.d. 
The  principle  here  established  is  the  distributive  principle 
of  multiplication  ;  and  theor.  7  is  sometimes  written,  multi- 
plication is  distributive  as  to  addition. 

But  addition  is  not  distributive  as  to  multiplication. 
E.g.,  (3  +  2)  X  5  =  (3x5) +  (2x5);  (3  x  2) +  5^(3 +  5)  x  (2 +  5). 

OPPOSITES. 

Two  numbers  whose  sum  is  0  are  ojyposites  of  each  other. 
E.g., +4, -4;  -2/3,  +2/3;  "3  x  "4,-12;  3+4,-7;  3-4,4-3„ 
Theor.  8.   The  sum  of  the  opposites  of  two  or  more  abstract 
numbers  is  the  opposite  of  their  sum. 
Let  +rt,  "^Z»,  •  •  •,  "m,  -7i,  •  •  •,  +p>/q,  —r/s,'  •  •,  be  any  abstract 

numbers, 
then  is  the  sum  ~a+-b-\ —  •  +  ^m  +  +  w  +  •  •  •  —p/q  +  r/s •  •  • 
the  opposite  of  the  sum 
+05++^+..-  +-m+~n+'  •  .  +p/q-r/s"- 


7, 8,  §4]  ADDITION  AND  SUBTRACTION.  27 

For  the  sum  of  these  two  sums 

+  •  •  .  +  (i?/^  -p/q)  4-  ( -  r/5  +  r/5) 
=  0  +  0+-      =0.  Q.E.D.         [df.  opp.,  th.  5. 

QUESTIONS. 

1.  In  the  proof  of  theor.  7  how  many  abstract  numbers  are 
used  ?  are  these  numbers  integers  or  fractions  ?  are  they  pos- 
itive or  negative  ?  is  m  abstract  or  concrete  ? 

2.  By  what  principle  are  the  two  products  equal : 

By  what  principle  is  the  second  product  expanded  into 
m  X  «  +  ??i  X  ~  ^  +  ??i  X  c  +  •  •  •  +  ?/i  X  li/k  +  711x1? 

By  what,  is  this  last  product  changed  to  the  form  sought  ? 

3.  Find  the  product  of  the  two  polynomials  a  —  b  +  c-\ —  • 
—  h/k  +  l  and  m  +  l/?i— J9,  and  show  that  their  product  is  the 
sum  of  the  partial  products  got  by  multiplying  each  term  of 
the  multiplicand  in  turn  by  each  term  of  the  multiplier. 

Does  the  order  in  which  the  terms  are  multiplied  affect  the 
final  product  ? 

4.  Find  the  product  ota  +  b,  c-\-  d,  e  +/,  and  show  that  each 
term  of  this  product  contains  three  letters:  one  from  the  first 
factor,  one  from  the  second,  and  one  from  the  third. 

So,  for  the  three  factors  a  —  h,  c  —  d,  e  — /. 

5.  In  the  case  of  two  opposite  numbers,  what  is  true  of  the 
two  measuring  units?  of  the  number  of  times  the  unit  is 
repeated  in  each  of  them?  of  the  quality  of  the  numbers  as  to 
that  of  the  unit?  of  the  tensors  in  the  abstract  operators? 
of  the  versors  ? 

6.  What  number  is  its  own  opposite  ? 

7.  How  is  the  sum  of  the  opposites  of  two  or  more  numbers 
related  to  the  sum  of  the  numbers  ?  the  product  of  the  oppo- 
sites of  two  numbers  ?  that  of  the  opposites  of  three  numbers  ? 

8.  Explain  the  dependence  of  theor.  8  on  theor.  5. 


28  THE  PRIMARY  OPERATIONS  OF  ARITHMETIC.     [I,  Ta 

SUBTRACTIOJq". 

Subtractio:n"  is  an  operation  that  is  the  inverse  of  addition, 
i.e.,  when  the  sum  of  two  numbers,  and  one  of  them,  are  given, 
subtraction  consists  in  finding  the  other  number.  The  sum 
is  now  called  the  7nmiiend,  the  given  number  is  the  subtra- 
hend, and  the  number  sought  is  the  remainder. 

The  sign  of  subtraction  is  — ,  read  minus  and  the  order  of 
writing  is  minuend  — subtrahend  =  remainder;  e.g.. 


$50- 

$40  = 

$10, 

for    $40  + 

$10  = 

$50, 

+  50- +40= +10 

$40- 

$50  = 

-$10, 

for    $50  4- 

-$10  = 

$40, 

+40- +50= -10 

-$50- 

-$40  = 

-$10, 

for-$40  + 

-$10  = 

-$50, 

-50--40=-10 

-$40- 

-$50  = 

$10, 

for -$50  + 

$10  = 

-$40, 

-40--50=+10 

$40- 

-$50  = 

$90, 

for -$50  + 

$90  = 

$40, 

+40- -50= +90 

-$50- 

$40  = 

-$90, 

for  $40+- 

-$90  = 

-$50, 

-50- +40= -90. 

Theor.  9.  If  to  the  sum  of  tivo  numbers  the  opposite  of  one 
of  them  be  added,  the  result  is  the  other  number. 
For  let  r,  s  stand  for  the  two  numbers  and  m  for  their  sum, 
then  •.*  m  =  r  +  5,  [^JP* 

.*.  ?«  +  ( —  6-)  =  r  +  5  +  ( —  s)  =  r  +  (5  —  5)  =  r,         q.e.d. 

Theor.  9  may  be  written,  the  sum  of  the  minuend  and  the 
opjyosite  of  the  subtrahend  is  the  remainder. 

Hence  the  terms  of  an  expression  enclosed  in  a  parenthesis 
and  preceded  by  a  minus  sign  can  be  added  to  like  terms,  pro- 
vided the  sign  of  every  term  so  enclosed  be  first  reversed. 

Cor.  If  there  be  a  series  of  additions  and  subtract io7is,  the 
final  result  is  the  same,  in  whatever  order  they  be  performed, 
a7id  however  the  elements  be  grouped. 

For  every  subtraction  is  an  addition  of  the  opposite  of  the 
subtrahend,  and  these  additions  may  be  performed  in 
any  order  and  the  terms  be  grouped  in  any  way. 

In  applying  this  corollary  the  pupil  must  take  care  lest  he 
change  the  office  of  any  element  and  add  it  where  he  should 
subtract  or  subtract  it  where  he  should  add. 
E.g.,  12  -  3  +  2  +  4  -  8  =  (12  -  3  +  4)  +  (2  -  8) 
¥=(4 -3 +  12) -(8 +2). 


9,  §4]  ADDITION    AND   SUBTRACTION.  22 

QUESTIONS. 

1.  May  the  three  numbers  involved  in  the  process  of  sub- 
traction be  all  abstract  ?  all  concrete  ?  part  abstract  and  part 
concrete  ? 

If  all  be  concrete  what  else  must  be  true  of  them  ? 

2.  Interpret  the  examples  given  under  the  definition  of 
subtraction,  when  the  positive  numbers  stand  for  assets  or 
earnings  and  the  negative  numbers  for  debts  and  expenses, 
and  show  what  is  meant  by  subtracting  a  negative. 

3.  In  arithmetic,  how  does  the  remainder  compare  in  size 
with  the  minuend?  is  this  always  true  in  algebra  ? 

4.  If  from  a  —  b,  c  +  d  was  to  be  subtracted,  and  c  alone  has 
been  subtracted,  is  d  to  be  added  to  this  remainder  or  sub- 
tracted from  it? 

Prove  that  [a  —  i)  —  {c  —  d  -\-  e)  —  a  —  Ij  —  c  -\-  d  —  e. 

5.  If  a  polynomial  be  enclosed  in  a  parenthesis  and  preceded 
by  a  minus  sign,  what  changes  must  be  made  in  removing  the 
parenthesis  ? 

So,  if  a  parenthesis  is  to  be  inserted  after  a  minus  sign,  what 
changes  must  be  made  in  the  signs  of  the  terms  included  in  it  ? 

6.  Simplify  3rt- (a- 4$ +  2«). 

8.  The  rule  for  algebraic  subtraction  is :  to  each  term  of  the 
minuend  add  the  opposite  of  the  like  term  of  the  subtrahend; 
what  is  the  origin  of  this  rule  ? 

9.  From  the  remainder  subtract  the  opposite  of  the  subtra- 
hend; what  is  found  ? 

What  other  operation  will  give  the  same  result  ? 

10.  What  two  processes  have  been  proved  commutative  and 
associative  ? 

What  other  two  processes  may  always  be  so  indicated  as  to 
be  examples  of  these  two?  with  what  caution  ? 

11.  Show  in  what  respects  theors.  4  and  9  are  alike  and  in 
what  they  differ. 


iiO         THE  PRIMARY  OPERATIONS  OF  ARITHMETIC.     [I,Th. 

§5.     INVOLUTION  AND  EVOLUTION. 

The  continued  product  of  a  number  by  itself  is  a  power  of 
that  number.  The  number  whose  power  is  sought  is  the  base, 
and  the  operator  that  shows  how  many  times  the  base  is  used 
as  a  factor  is  the  expo7ient ;  it  is  written  at  the  right  and  above 
the  base.  Involution  is  the  process  of  finding  powers. 
E.g.,  4x4x4  =  4^=64,  1/4x1/4x1/4  =  1/64.      [th.  S,  cr.  2. 

In  both  examples  the  base  is  4;  in  the  first  the  operation  is 
a  continued  repetition  by  4,  and  in  the  other  it  is  a  continued 
partition  by  4,  operations  that  tend  to  neutralize  each  other; 
and  this  relation  may  be  expressed  by  writing  them  4+^  and  4-^, 
wherein  the  positive  exponent  shows  how  many  times  4  is  used 
in  repetition  and  the  negative  exponent  shows  how  many 
times  4  is  used  in  partition. 

As  an  exponent,  "1  reverses  the  quality  of  the  base,  i.e.,  if 
the  base  denote  repetition,  the  exponent  ~1  changes  the  opera- 
tion to  partition,  so  that  n~'^  —  l/n. 

The  words  positive  and  negative  as  applied  to  powers  refer 
to  the  exponents  only;  and  integer  potuers  are  powers  whose 
exponents  are  integers. 
E.g.,  (~4)^  is  a  positive  integer  power,  although  its  value,  "64, 

is  negative. 
So,  4~^  is  a  negative  integer  power,  although  its  value,  1/64,  is 
positive  and  a  fraction. 

PRODUCT  OF   INTEGER   POWERS   OF   THE   SAME   BASE. 

Theor.  10.  The  product  of  tivo  or  more  integer  powers  of  a 
base  is  i}iat  power  of  the  base  luhose  exponent  is  the  sum  of  the 
exponents  of  the  factors. 

Let  A  be  any  number,  I,  m,  n,  •  •  -any  positive  integers  ; 
then  the  product  A^  x  A"*  x  A""  '  * '  is  a' +"»-'*  •  •  • 
For  *.•  a'  =  A  X  A  X  •  •  •/  times,  A'"  =  A  x  A  x  •  •  •■?y^  times, 

a~"  =  1/a  X  1/a  X  •  '  -n  times  [df.  int.  pwr. 

/.  a'  X  A"*  X  A""  -  (a  X  A  X  .  • .  l  +  7n  - n  times)  x  (a  x  1/a) 
X  (a  X  1/a)  X  ' '  '71  times,  when  l  +  7n^7i. 


10,  §5]  INVOLUTION   AND   EVOLUTION.  31 

and  •/  ax1/a  =  1, 

.-.  a^xa'"xa-"  =  a'+"*-"; 


aud      a'x  A*"x  a-"=(1/ax  1/a  x  •  •  -n  —  l+m  times) 

(ax1/a)  X  •  •  •  l-\-m  times  when  l-\-m<  /^  =  1/a'*"^"'" 
—  ^i+m-n.^  Q.E.D.         [df.  neg.  pwr. 

aud  so  for  more  than  three  factors. 

Cor.  Tlie  quotient  of  two  integer  powers  of  the  same  base  is 
that  power  of  the  base  whose  exponent  is  the  exponent  of  the 
dividend  less  the  exponent  of  the  divisor. 


QUESTIONS. 

1.  By  diagram  show  the  square  and  the  cube  of  2. 

Can  the  higher  powers  of  2  be  represented  by  diagrams  ? 

2.  Can  a  concrete  number  be  raised  to  a  power? 

3.  Of  what  numbers  are  the  high  powers  larger  than  the 
low  powers  ?  smaller  ?  the  same  ? 

4.  What  is  the  value  of  (  —  2)*  and  what  kind  of  power  is  it  ? 
So,  of  (  +  2)-^  of  (-if,  of  5^  of  5-3? 

5.  What  are  the  products  S^x  5"^?  {-^Y>^{-h)'\  a^xa-^^ 

6.  What  power  of  b  is  the  product  b"^  x  b''  x  (l/Z/)'"  ?    How 
many  of  the  factors  b  are  cancelled  ?  how  many  remain  ? 

So,  of  the  products  b"^  x  b''  x  {l/bf-,  b"^  x  b''  x  [l/bf  x  {l/by  ? 

7.  What  relation  has  {l/by  to  b-"^? 

8.  What  relation  has  the  quotient  b"^  :  J"  to  the  products 
^>"*x^-"and  ^'"x(l/Z>)"? 

9.  How  many  times  is  &  a  factor  of  the  product  b^xb"^? 
How  many  of  these  factors  can  be  cancelled  by  the  factors 

of  5~"  ?  how  many  cannot  be  so  cancelled  ? 

10.  Granted  that  b^  xb""  xb-''  =  b^+"'-'',  what  is  true  of  the 
exponent  of  the  product  of  integer  powers  of  a  number? 

11.  Divide  a'  by  a%     x'  by  x-%     Z*-^  by  b^     m-^  by  ifi'^ 


For 

(A"-)" 

So, 

(A-)- 

So, 

(A-™)' 

So, 

(A-) 

32  THE   PRIMARY   OPERATIONS   OF   ARITHMETIC.  [I,Ths. 

INTEGER    POWER   OF    AN    INTEGER    POWER. 

Theor.  11.  An  integer  power  of  an  integer  power  of  a  base 
is  that  power  of  the  base  whose  exponent  is  the  p^roduct  of  the 
two  given  exponents. 

Let  A  be  any  number;  m,  n  any  positive  integers; 
then  (a^)"i=a'«^ 
"  =  A*"  X  A"*  X  •  •  •  w  times 

_^m  +  m-f-...ntime8_^mxn^  Q.E.D.  [th.   10. 

=  I/a""  X  I/a''*  X  '  '  -n  times 

1  /  A  w*+  m  -t- .  . .  n  times i  /  k^^^^ a  —inn 

=  A~"*xA~*"x  •  •  -n  times 

--.i-m-m-...n  times »  -  mn 

'^I/a-^'xI/a-^^x  •  • -^i  times 
=  A"*  X  A*"  X  •  •  •  ?i  times  =  a"*". 

PRODUCT   OF    LIKE    INTEGER    POWERS   OF    DIFFERENT   BASES. 

Theor.  12.  The  product  of  like  integer  poivers  of  tivo  or 
more  bases  is  the  like  poiver  of  the  product  of  the  bases. 
Let  A,  B,  1/c  be  any  numbers  and  n  any  positive  integer; 
then  A"  X  B"  X  1/c"  =  (a  x  b/c)". 

For  •.•  A"  =  A  X  A  X  • '  -n  times,  b"  =  b  x  b  x  -  •  -  71  times, 
1/c"  =  1/c  X  1/c  X  ' ' '  n  times, 

,•.  A"  X  B"  X  1/C"=  (a  X  B  X  1/c)  X  (a  X  B  X  1/c)  X  •  •  • 

n  times  =  (a  x  b  x  1/c)".  q.e.d.         [ths.  1,  2. 

So,   *.*  A""=1/a  X  1/a  X  '  '  -n  times,  b~"=  1/b  x  1/b  x  •  •  • 
n  times,  l/c~"  =  c  x  c  x  •  •  •  ;i  times, 
/.  a-"xb-"x1/c-"=:(1/ax1/bxc)  X'(1/ax1/bxc)-  •• 
n  times  =  (c/a/b)"  =  (ax  b/c)-".  q.e.d. 

Cor.  The  quotient  of  two  like  integer  powers  of  different 
bases  is  the  same  potver  of  the  qiiotient  of  the  bases, 

EVOLUTION. 

Evolution  is  an  operation  that  is  tlie  inverse  of  involution; 
i.e.,  it  consists  in  finding  a  base  tiiat,  raised  to  a  given  power, 
is  a  given  number.    It  is  a  process  of  trial  and  test.    The  base 


11,12,13,151  INVOLUTION   AND   EVOLUTION.  33 

is  now  called  the  root,  and  the  exponent  of  the  power  is  the 
roof-index.     The  radical  sign,  /^,  is  placed  before  the  number 
whose  root  is  sougiit,  and  the  root-index  at  the  left  and  above 
this  sign;  the  root-index  2  need  not  be  written. 
E.g.,    f4,  orsimply  |/4,  =  +2,  or-2;     |/343  =  7j    |/343=:-7. 

Evolution  being  the  inverse  of  involution,  the  following 
converse  theorem  follows  directly. 

Theor.  13.  An  integer  root  of  an  integer  jioiver  of  a  hase  is 
that  power  of  the  base  luhose  exponent  is  the  integer  quotient 
of  the  exponent  of  the  power  ly  that  of  the  root. 
E.g.,     y'A'^^^A". 

QUESTIONS. 

1.  What  cases  are  considered  in  theor.  11  besides  a  positive 
integer  power  of  a  positive  integer  power  ? 

2.  Write     (a""*)""     so  that  positive  exponents  only  shall  be 
used;  then,  so  as  to  express  the  result  as  a  power  of  A. 

3.  In  the  product     A"  x  b"  x  l/c'*,     how  many  times  is  A  a 
factor?  B  ?  1/c  ?  how  many  factors  in  the  entire  product? 

Into  how  many  groups  of  the  form     a  x  b  x  1/c     can  these 
factors  be  divided  ?   Indicate  this  grouping  in  the  simplest  way. 

4.  Why  does     l/c""  =  c  x  c  x  c  •  •  •  w  times  ? 

5.  Write     A""  x  B""  x  (1/c)-"     with  positive  exponents. 
How  many  times  is  the  group     1/a  x  1/b  x  c    a  factor  of 

this  product  ?  what  power  is  it  of  the  fraction     A  x  b/c  ? 
G.   Prove  that     a'Vb'"  =  (a/b)'". 

7.  What  powers  of  numbers  and  of  their  opposites  are  the 
same?  not  the  same  ? 

8.  To  what  element  in  involution  does  a  root  correspond  ? 
the  root-index?  the  number  whose  root  is  sought  ? 

9.  What  number  has  the  same  effect  whether  used  as  a 
root-index  or  as  an  exponent? 

10.  |/4=  +2,    t^4=  -2:    is  +2,  or  "2,  the  square  root  of  "4? 

11.  Find  the  values  of     {x^f,     {i/-')\     {h')-\     (rt-^)"*. 

12.  Find  the  values  of     ^x\     l/ij-\     ^b-^\     i^a^K 


34         THE   PRIMARY   OPERATIONS  OF   ARITHMETIC.  [I. 

§  6.    QUESTIONS  FOR  REVIEW. 

Define  and  illustrate : 

1.  Quantity;  unit;  unity;  number;  integer;  fraction. 

2.  Concrete  numbers;  equal  concrete  numbers. 

3.  Abstract  numbers;  equal  abstract  numbers. 

4.  Product;  multiplication;  division;  quotient;  the  prod- 
uct of  two  abstract  numbers. 

5.  Positive  numbers;  negative  numbers. 

6.  Sum;  addition;  subtraction;   remainder;   the  sum  of 
two  abstract  numbers. 

7.  Reciprocals;  opposites;  a  number  larger  than  another. 

8.  Power;  involution;  evolution;  root. 
State  and  prove : 

9.  The  associative  principle  of  multiplication. 

10.  The  commutative  principle  of  multiplication,  four  cases. 

11.  The  associative  and  commutative  principles  of  addition. 

12.  The  distributive  principle  of  multiplication. 

13.  The  principle  on  which  the  theory  of  division  rests. 

14.  The  principle  on  which  the  theory  of  subtraction  rests. 

15.  The  principle  by  which  the   product  of  two  integer 
powers  of  the  same  base  is  found. 

How  does  this  principle  apply  to  their  quotient? 

16.  The  principle  by  which  an  integer  power  of  an  integer 
power  is  found. 

How  does  this  principle  apply  to  the  extraction  of  roots? 

17.  The  principle  by   which   the  product  of  like  integer 
powers  of  different  bases  is  found. 

How  does  this  principle  apply  to  their  quotient? 

18.  What  two  offices  has  an  abstract  negative  multiplier? 

19.  How  can  a  series  of  additions  and  subtractions  be  made  ? 
a  series  of  multiplications  and  divisions?  with  what  caution  ? 

20.  What  is  the  product  of  the  reciprocals  of  two  or  more 
numbers?  the  sum  of  their  opposites? 


«6]  QUESTIONS  FOR  REVIEW.  35 

21.  What  operation  is  the  inverse  of  multiplication?  of 
addition?  of  involution?  What  inverses  has  the  multiplica- 
tion $5x4?  the  involution  4^? 

22.  Can  two  concrete  numbers  be  added  together  ?  with  what 
caution?    Can  one  such  number  be  subtracted  from  another? 

A  man  may  walk  a  mile,  then  take  another  step:  can  a  mile 
and  a  yard  be  added  together? 

23.  Can  two  concrete  numbers  be  multiplied  togetlier  ?  Can 
one  such  number  be  divided  by  another  ?  with  what  caution? 

24.  Can  a  concrete  number  be  raised  to  a  power  ?  Can  a 
root  be  taken  of  such  a  number  ? 

25.  What  is  the  meaning  of  a  concrete  fraction  ?  of  an  ab- 
stract fraction  ?   Are  both  terms  of  a  concrete  fraction  concrete  ? 

26.  What  effect  has  it  upon  a  fraction  if  both  terms  be  mul- 
tiplied by  the  same  number?  if  both  be  divided  by  the  same 
number?  if  the  same  number  be  added  to  both  terms?  if  both 
terms  be  raised  to  the  same  power  ?  if  like  roots  be  taken  ? 

27.  Group  the  factors  below  in  such  a  way  as  to  make  the 
multiplication  easiest;  hence  explain  cancellation  in  the  mul- 
tiplication of  fractions:     3/2- 7/1 1-25/51. 2/7- 11/5.  lT/5. 

28.  Is  involution  distributive  as  to  multiplication?  is  evolu- 
tion?    Are  these  two  statements  true  equations? 

(12  X  6)2=r  12^  X  6^     |/(27  x  8)  =  -^27  x  ^S. 

29.  Is  involution  distributive  as  to  addition  ?  is  evolution  ? 
Are  these  two  statements  true  equations  ? 

(12  +  6f  =  12H6^     ^{27  ±8)  =  ^27  ±^8. 

30.  Suggest  questions  that  could  not  possibly  be  answered 
by  a  negative  number;  not,  by  a  fraction. 

31.  Show  that  multiplication  maybe  defined  as  the  process 
of  doing  to  one  of  two  numbers  that  which,  when  done  to  a 
unit,  produces  the  other. 

32.  Why  is  it  that,  instead  of  dividing  by  a  composite  num- 
ber, one  may  divide  by  one  of  its  factors,  that  quotient  by 
another  factor,  and  so  on,  till  all  the  factors,  are  used  ? 

33.  Why  does  |/(108  x  27  x  250  x  490)  equal  6  x  9  x  5  x  70  ? 


36  THE   PRLMARY   OPERATIONS   OF   ALGEBRA.  [H, 


II.    THE  PRIMARY  OPERATIONS  OF  ALGEBRA. 


In  principle,  the  operations  of  algebra  differ  not  at  all  from 
the  like  operations  of  arithmetic  ;  the  only  differences  arise 
from  the  differences  between  the  forms  of  algebraic  expressions 
and  the  simpler  forms  common  in  arithmetic. 
E.g.,  the  sum  of  4  and  5  is  9,  and  their  product  is  20; 
but  the  sum  and  product  of  a  and  I  can  only  be  expressed  by 

writing  a-\-h  and  ay<h,  a-h,  ov  ab, 
which  mean  that  an  addition  and  a  multiplication  are  in- 
tended, and  that  they  will  be  effected  when  the  values 
of  a  and  b  are  made  known. 

§1.    ALGEBRAIC   EXPRESSIONS. 

An  algebraic  expression  is  a  number  or  combination  of 
numbers  written  in  algebraic  form.  It  is  called  an  expressio?i 
or  a  number^  according  as  the  thought  is  of  the  symbol  or  of 
the  value  that  the  symbol  represents. 

Unless  it  be  a  single  letter  or  numeral,  an  expression  is 
made  up  of  simpler  expressions  affected  or  combined  by  signs 
of  operation. 

The  parts  of  an  expression  that  are  joined  by  the  signs 
+  or  —  are  terms. 

An  expression  of  one  term  only  is  a  monomial,  of  two  terms 
a  binomial,  of  three  terms  a  trinomial,  of  four  terms  a  quadri- 
nomial,  of  two  or  more  terms  a  polynomial 

An  expression  is  numerical  if  tlie  numbers  be  expressed 
wholly  by  numerals,  literal  if  wholly  or  in  part  by  letters; 
finite  if  the  number  of  operations  implied  be  limited,  infinite 
if  unlimited. 

An  algebraic  expression  is  entire  if  it  be  free  from  divisors 
and  vooi^j  fractional  if  not  free  from  divisors. 


511  ALGEBRAIC   EXPRESSIONS.  37 

When  the  terms  of  an  expression  are  so  related  to  each 
other  that  each  successive  term  is  derivable  by  some  fixed  law 
from  the  previous  terms,  the  expression  is  a  series. 
E.g.,  l  +  x-\-x^-{-a?-\- ' ' '  +X''  h  Q.  finite  series,  if  r  be  any  given 

integer,  arranged  according  to  rising  poivers  of  x\ 
but    1  +  a;  +  o;^ + a,-^  +  •  •  •  +  re""  +  •  •  as  an  infinite  series, 

QUESTIOJ^S. 

1.  Is  d/l)  an  algebraic  expression  or  a  number? 
If  a  be  the  entire  cost  of  b  books,  what  is  a/h  ? 

2.  What  name  is  given  to  the  parts  a,  h,  of  the  expression 
a/b  ?  of  the  expressions    a  —  b,    a  +  b,     a-b,    a^,    ^a  ? 

3.  What  three  pairs  of  contrasting  names  are  applied  to 
algebraic  expressions?  define  and  illustrate  them. 

4.  In  the  series      l-\-x-\-x^-{- '  '  -  +x'',      how  is  the  third 
term  got  from  the  second  ?  the  fourth  from  the  third  ? 

If  r  be  7,  how  many  terms  are  in  this  series  ? 

How  many  operations  are  performed  ?  what  are  they  ? 

Is  the  series  finite  or  infinite? 

5.  In   the   series     l+x  +  x^-\ +x''-\ ,     how   many 

terms  are  there  after  of  ?     What  is  the  twelfth  term  of  this 
series?  the  twentieth?  the  ?itli? 

6.  It  a  =  4:,  b  =  l,  c  =  2,  d=9,  a;  =  5,  y  —  8,  find  the  values  of 

V8r/c,    7^5dx,    a^-W-'6abCy    sly"",    sjb^,    {2d-5c)\ 

7.  It  a  =  If     b=—3,     c  =  5f    find  the  value  of 
a^b^  +  1     l-^/,V     2b'' -^ac     a^  +  2ab  +  I^ 


+  ■ 


a^^b^       a^-c^^    b^-c"       b''-2bc  +  c^' 

8.  If  a  =  25,   b^.9,    c=-4,    d= -I,  find  the  value  of 

sj  -bc-VZsJacd-^sl  -Wd^-sl  -(?d, 

9.  If  a  =  0,   b=-2,   c  =  4,   d--^,   find  the  value  of 

3  |/(2Z>2-«)4-2  ^(^^Hc^  +  7)-  ^/\2{b■^cf^  (d^bY^bc\ 

10.  If  a;=  1/2,  how  much  will  the  sum  of  the  first  three  terms 
of  the  series  l+a:  +  a:^  +  ^+  •  •  •  lack  of  2? 

So,  the  first  four  terms?  five  terms?  ten  terms?  n  terms? 


38  THE  PRIMARY  OPERATIONS  OF  ALGEBRA.  [11, 

SYMMETRY. 

As  to  any  of  its  letters,  an  expression  is  symmetric  when  its 
value  is  unchanged  however  those  letters  exchange  places. 
E.g.,  the  expressions  xxyxz  and  x-\-y  +  z  are  symmetric  as 

to  X,  y,  z,  or  as  to  any  two  of  them. 
So,  io-\-x  —  y  —  z  is  symmetric  as  to  w,  x,  and  as  to  y,  z;  but  not 

as  to  2U,  y,  as  to  w,  2,  as  to  x,  y,  or  as  to  x,  z. 

DEGREE — HOMOGENEOUS  TERMS. 

The  sum  of  the  exponents  in  a  monomial  is  its  degree. 

The  degree  of  a  polynomial  is  that  of  the  term  whose  degree 

is  highest  of  all. 

A  polynomial  made  up  of  terms  all  of  the  same  degree  is 

liomogeneous.     Expressions  having  the  same  degree  are  hoyno- 

geneotts  loith  each  other. 

E.g.,  a^  +  3a'^  +  3«^'  +  h^  is  homogeneous  and  of  the  third 
degree. 

So,  rt%  rt"-'^,  a"-'^',  •  •  •  ft"-^^'*,  .  •  •  alf-\  }f  are  homogeneous 
with  each  other  and  of  the  nW\  degree. 

And  ax^y   h'^xy,   c^y""  are  of  the   second  degree,  and  homo- 
geneous with  each  other  as  to  a:  and  y, 

but  of  the  third,  fourth,  fifth  degrees,  and  not  homogeneous, 
as  to  all  the  letters. 

COEFFICIENTS — LIKE  AND   UNLIKE  TERMS. 

If  an  expression  have  a  numerical  factor,  then  usually  the 
numeral  alone,  together  with  the  sign  of  the  number,  +  or 
— ,  or  the  numeral  and  one  or  more  of  the  letters  that  follow 
it,  is  called  the  coefficient. 
E.g.,  in  labc,  7  is  the  coefficient  of  abc,  Ha  of  be,  lab  of  c. 

Terms  that  differ  only  in  their  coefficients  are  like  (similar) 
terms;  other  terms  are  unlHiC. 
E.g.,  baXj  lax  are  like;  but  5«:r,  Iby  are  unlike. 
So,  bax,  Ibx  are  like  if  5r?,  lb  be  counted  as  the  coefficients 
of  X ;  but  unlike  if  5,  7  be  coefficients  of  ax^  bx. 


§1]  ALGEBRAIC  EXPRESSIONS.  39 

QUESTIONS. 

1.  If  a:  =  3,  y  =  3,  2  =  4,  w  —  6,  is  the  expression  xxyxz 
the  same  as  2x3x4?  as  3x2x4?  4x2x3?  2x4x3? 
3x4x2?     4x3x2? 

So,  is  wi-x-y-z  thesameas  6  +  2-3-4?  4  +  6-3-2? 
2  +  6-4-3?  6  +  3-2-4? 

2.  In  the  expression  a  x  b  x  c  x  d,  what  letters  may  be  in- 
terchanged without  changing  the  value  ? 

So,  in  the  expression    a  +  b  —  c-{'d?    in  ab/cd?  in  ab  +  cd? 

3.  What  is  the  degree  of  a  ?     of  a^  ?    of  ab  ? 

4.  Of  the  trinomial  a^  —  abc  +  c,  which  term  is  of  the 
highest  degree  ? 

What  is  the  degree  of  this  expression?  is  it  homogeneous  ? 

Supply  the  terms  that  are  wanting  so  as  to  complete  the 
homogeneous  series: 

5.  a'-\-a'b  +  a'b^i-"-i-ab'  +  b\ 

6.  a' -\-a'b  +  a'b^+  •  "  +ab'  +  b\ 

7.  a'-a'b-ha'b^ ab^  +  b\ 

8.  a'~a'b  +  a'b' +ab'-b\ 

9.  «"  +  a"-i^  +  a"-2^2^  . . .  +ab''-^  +  b\ 

10.  «"-«^"-ii  +  a"-2^2 ab''-^  +  b\ 

11.  a''-a''-''b  +  ar  ^V +a5"-^- J". 

12.  If  n  be  even  will  the  series  a^  —  d^~^b-\ —  •  end  with 
-b""  or  with  +^>"  ?  if  n  be  odd  ? 

13.  Supply  the  terms  that  are  wanting  and  complete  the 
symmetric  series     a^  +  a^b  +  d^c-{ —  •  +  abc  H +6-^. 

14.  So,     a{x^  +  ?/'  +  2^)  +  b(x^i/  +  x^z+'")-\- cxyz. 

15.  If  an  expression  be  symmetric  as  to  every  pair  of  letters 
involved,  it  is  symmetric  as  to  all  the  letters,  and  conversely. 

16.  In  the  product  ^a^bc^  what  is  the  coefficient  of  be"  ?  of  (;^? 

17.  Are3rt:r,     ba%     -^bx,     |/At,     like  terms  ? 
Can  they  be  so  taken  as  to  be  made  like  terms  ? 

18.  As  an  operator,  what  process  does  a  coefficient  indicate? 


40  THE  PRIMARY   OPERATIONS  OF   ALGEBRA.     [n,PRs. 

§2.  ADDITION   AND   SUBTRACTION. 

PrOB.   1.    To   ADD    TWO    OR    MORE    NUMBERS. 

(«)  The  nu7nhers  like:  to  the  common  factor,  prefix  the  sum  of 
the  coefficients.  [I  th.  7. 

E.g.,  10  ft.  down +  20  ft.  up     +60  ft.  up       =70  ft.  up, 

10  ft.  up      +20  ft.  down +  60  ft.  down  =  70  ft.  down. 
So,     10.r  -  15x  +  20a:  -  25x  +  ZOx  =  60.r  -  40.?;  =  20x, 

lOay  +  20by  -  SOci/  =  {lOa  +  20^  -  306)?/. 

(b)  The  numbers  unlike:  write  the  numbers   in  succession, 

tvith  their  signs,  in  any  co7ivenient  order,         [I  th.  5. 
E.g.,  19a:?/2;  — 29m?i  +  39rt  — 49  is  irreducible. 
So,     lOay  -\-20by  —  dOcy  is  usually  not  reduced,  but  may  be 

written  {10a-h20b-'30c)y,  as  above. 

(c)  Some  numbers  like  and  some  unlike:  unite  into  one  sum 

each  set  of  like  numbers  ; 
write  these  partial  S2cms,  together  with  the  remaining  terms, 

in  any  convenient  order. 
E.  g. ,  (a^  +  Sa^^  +  3«^>H  ^')  +  («'  -  '^ah  +  ^ao^  -  c') 
=  2a''  +  ?>a\b-c)  +  '^a{W  +  c^)-^{b^-o^). 

SUBTRACTION". 
PrOB.  2.    To    SUBTRACT   ONE    NUMBER   FROM    ANOTHER. 

By  trial,  or  memory,  find  what  nmnber  added  to  the  sub- 
trahend  gives  the  minuend.  [df.  sub.,  p.  28. 

Or,  to  the  minuend  add  the  opposite  of  the  subtrahend, 

[I  th.  9. 
In  general,  the  first  rule  is  best  when  the  numbers  are  like, 

and  the  second,  when  tliey  are  unlike. 

E.g.,  7rt  —  4rt  =  Sry,         la  -  -'ia  —  \0a. 

So,       -la-'^Sa-'lOa,         -7rt- -3r/= -4«. 

So,      la  --Ab^la  +  4b,         la  -  '  4//=  la  -  4  J. 

So,  [2^3  +  ^a\b  -  c)  +  3^/(/^2  +  r)  +  {U'  -  c')]  -  [«=»  -  'ia'^c  +  Zac^  -  c^] 
=  a^  +  Wb^-'^ab''^b\ 


dd 

QUESTIONS. 

1. 

7x, 

15.r,     -3x, 

-8x, 

2x,     12x, 

~l0Xf 

272-, 

2. 

da% 

ISa'y,     - 

'^bahj. 

-42«^?/, 

Z'dahj, 

2ba'^y, 

1,2,  §2]  ADDITION  AND  SUBTRACTION.  41 


\lx. 


3.  Iiax^y     -4:a^x,     oa^x,     9a^x,     ~15ax^,     ~26a^x, 

4.  7a-{-bc-dx^i/,     6x^i/-8b-4:a,     U  +  4:X^ij  +  2c, 

5.  Smii'  +  ^x'f  +  ba,    Ixhf  +  '^mii'-la,    2mn^-\l3^tf  +  2(u 

6.  (^-2;;^,    {q-hV^    (3c-2r)a:,    {^p-a)o?,    ~{h  +  q)A 
—  {p  —  (()'^f  [arrange  the  sum  to  rising  powers  of  x. 

7.  a'  -  a'b  +  a'b^  -  . .  »  -  ab^,     a^b  -  a'b^  +  aW b\ 

8.  a^ - a%  +  en'' +  ab\     a^b - a^'b^  +  a'b^ +b\ 

10.  From  Ix  subtract  ox,  bx,  '7Xj  9x,  II.t,  in  turn,  and  add 
the  five  remainders. 

11.  So,  from  13^y  subtract  bay^,  dai/^,  ay^,  —aif,    —  3r/?/^ 
Find  the  value  of 

12.  («  +  ^  +  c-2+5.r)-(rt-J-9  +  13.r)  +  (rt  +  Z'-c  +  8.?;). 

13.  (7  +  5Z^)-[(5«.2;  +  3^)-2)-(4;?2  +  3«.^-4i)]. 

14.  {8xy  +  \\:>9ifz  -  80? yz*)  -  ( 2>xy  -  83? yh  -  '^x^yz*'). 
Free  from  brackets  and  reduce  to  the  simplest  form 

(fl)  removing  first  the  inner  brackets,  and  going  outwards; 
{b)  removing  first  the  outer  brackets,  and  going  inwards; 
(c)  freeing  all  terms  of  a  kind,  from  all  the  brackets: 

15.  a-lb-ie-d)].   16.  a-\a  +  b-[a  +  b-c-(a-h  +  c)']], 

17.  -{(l  +  2a;  +  9:^*2)  +  [(3  +  2.r-.7^2)-(-3  +  3.r-22:«)]}. 
Introduce  brackets,  taking  the  terms  two  together  in  their 

present  order,  and  having 

(a)  each  bracket  preceded  by  a  plus  sign; 

(b)  each  bracket  preceded  by  a  minus  sign; 
{()  the  first  term  in  each  bracket  positive: 

18.  -3c  +  4f?-2e  +  2/+2rt-5^. 

19.  a-^b  +  c  —  a  —  b  +  c-Va  —  b  —  c  —  a  +  h  —  c. 

30.  abc — abd  +  abe  —  acd + ace  +  ade  —  bed + bee — hde + cde. 


42  THE  PRIMARY  OPERATIONS  OF  ALGEBRA.      [II,  Pr. 

§  3.  MULTIPLICATION. 

PrOB.   3.   To   MULTIPLY   ONE   NUMBER   BY    ANOTHERc 

(a)  A  monomial  by  a  monomial:  to  the ^iro duct  of  the  numer- 

ical coefficients,  annex  the  several  letters,  each  taken  as 
many  times  as  it  ajopears  in  both  factors  together  ; 

[I  ths.  1,  2,  10. 

mark  the  'product  positive  if  the  factors  be  both  j^ositive  or 

both  negative,  and  negative  if  one  factor  be  positive  and 

the  other  negative.  [df.  ^\Qg.  oper.,  p.  20. 

E.g.,  +9«^»-^x  +7«2=  +63rt3Z>-^       -bxz-^x^lxh-^::^  ~Zbx'z-\ 

+  9a-^^x-'7a^:^-Q3a-''P,      -5xz-^x-'7x-^z^=+3bx-\ 

(b)  A  polynomial  by  a  monomial:  tmdtijjly  each  term  of  the 

multiplicand  by  the  multiplier  ; 
add  the  partial  products,  [I  th.  7. 

E.g.,  {?>xy^-\x-^z)  X  -^xy-h^zzz  -^fx^y-h^  +  ^x-^y-h\ 

(c)  A  polynomial  by  a  p)olynomial:  multiply  each  term  of  the 

multiplica7id  by  each  term  of  the  multijjlier; 
add  the  partial  products.  [I  th.  7  cr. 

E.g.,  (rt^  -ab^  W)  X  («  +  ^)  =  fl3  -  a%  +  ab^  +  a'^b  -  ah^  +  b"" 
^a^^bK 
The  work  takes  this  form:  c^  —  ab  +&^ 

a  ^b 

a^  —  a^b  -\-ab^ 
+  a^-ab^-\-P 


a'  +b' 

So,  to  multiply  ax^"  +  2bxy  +  cif  +  2dxz-\-2eyz  -V  fz^ 
by  mx  +  7iy  -Vpz :     write 

aa;2  +  ibxy  +  cj/^  +  "Idxz  +  2e  j/z  +  /z^ 
mx  +  ny  +  pz. 

amx3+26m  j  xi^y+cni  xxij"^      ■\-'ii.hn    x^z+Sou    xijz  -\-fm      xz"^ 

-\-an   I       -\-'ibn  I     -{-cny^  -^-2dn       -\-'icn\  y-^z  +fn  I  yz^ 

+(,p  +-2hp       +cp  I      +2,1  p     +2ep\    -\-fpz: 

Checks.    The  work   is  tested   by  division  [prob.  4],  and 
often  by  the  principles  of  note  1  below. 


3,  §3]  MULTIPLICATION.  43 

QUESTIONS. 

1.  In  repetition  by  a  positive  multiplier,  what  relation  has 
the  sign  of  the  product  to  that  of  the  multiplicand  ?  in  par- 
tition ? 

2.  Of  what  two  factors  is  a  negative  multiplier  composed  ? 
Does  the  tensor  change  the  sign  of  the  multiplicand?  does 

the  versor? 

What  is  the  sign  of  the  product  of  a  positive  number  by  a 
negative  number  ?  of  a  negative  number  by  a  negative  number? 
Multiply 
3.  {a  +  b)x{c-d).  4.   {2a'b-c^  +  2x)x{?>ah-c  +  x). 

5.  {?>a^-2crb-ay  +  W)x{(r  +  2ah-V). 

6.  [x"  -  2xij  -f-  if)  X  (.^2  +  if)  X  (x^  +  2XIJ  +  if)  X  {x^  -  y% 

7.  5  {x^  +  x^y  +  x^y^  +  xy^  +  y^)  x  (:c  -y). 

8.  {4:c'd-':cd^-vU^)  X  {6(^  +  4:C^d-dcd'). 

10.  (Z;2-^>c  +  c2)x(JH^'C  +  c2)x(Z/  +  6-)x(Z>-c). 

11.  (oxy  -  Zxz  +  2yz)  x  (rt:c  -  2^y  +  36-^). 

12.  {^a-x-\-2)x{a-4:X-l).    13.  {x^-ax^  +  'iif)x{x-ay). 

14.  (Sa^a;  -  «a;  +  a;)  X  (2«  -  ^)  x  (3«  +  2). 

15.  (4a-3-2z-2  +  a;-i-6)x(32;2-«3). 

16.  (3«2  +  5a5  -  4«c  +  8^2  -  (?)  x{a-  2ic  +  Sc^). 

17.  [px''y-{h-?,c)]x{l)-l)x{x-Vc), 

18.  (2.T?/  +  :^-3/)x(3:c2_^^_^2/). 

19.  (a*-2ftH3ff2-2r^  +  l)x(«*  +  2rt^  +  3ft2  +  2«  +  l). 

20.  (5a;H4:?:2_3_7^-i^;L0.i'-3)x(arHa;-2). 

21.  (17«'^.T-i  +  12rt&-V-9^6-2)  x(5J22:  +  6«-iZ*-i). 

22.  (3«/  +  4a2?/-6-10^-V-i-5?/-2)x(/-^%). 

23.  (i:z^-a;2  +  |.T-2)x(2a.'2  +  fT2  +  :^4-i). 

24.  {x^  +  2x-'^)x{x^-x  +  \).     25.  [a;^  -  (^>  +  c)a;  +  Jc]  x  (.t  -  «). 

26.  [2;H«'-«a;(a;3  +  «')J  x  \x^ ^- a^ - ax(x  +  a)\ 

27.  (a:'-3a;'  +  3a;-l)x(a;2  +  2A'  +  l)x(a;  +  l). 


44      THE  PRIMARY  OPERATIONS  OF  ALGEBRA.   [II.Pr. 
FORM  OF  PRODUCT. 

Note  1.  Certain  general  principles  are  manifest: 

1.  The  form  of  a  product  is  i7ide2)endent  of  the  values  of  the 
letters  that  enter  into  it, 

'E.g.,  {a  +  b)x  {a  —  b)  =  a'^  —  b^,  whatever  be  the  values  of  a,  b, 

2.  If  each  factor  be  symmetric  as  to  ttoo  or  more  letters,  the 
product  is  also  symmetric  as  to  the  same  letters. 

E.  g.,  (r/2  +  ^ab  +  b^)  x  («  +  ^)  =  a^  +  3a^  i-Sab^  +  b\ 

3.  If  any  values  be  given  to  the  letters,  the  value  of  the 
product  equals  the  product  of  the  values  of  the  factors. 

E.g.,  if  rt  =  5,  Z>  =  2,  then  ^H  2^^  +  ^2  =  25 +  20 +  4  =  49,  [above. 
«  +  ^=:5  +  2  =  7,     rtH3rtH3rt^H^>'  =  49-7  =  343. 

4.  57^6  sum  of  the  coefficients  of  a  ^n^oduct  is  the  continued 
2)roduct  of  the  sum  of  the  coefficients  of  the  first  factor,  by  the 
sum  of  the  coefficients  of  the  second  factor,  and  so  on. 

E.g.,  (4^  +  U)  X  (5«  -  U)  x{a~-  3b)  =  20a^  -  b3a^  -  27 ab^  +  I8b\ 
(4  +  3)  X  (5 -2)  X  (1-3)  =  20 -53 -27  + 18= -42. 

5.  The  degree  of  the  highest  teryn  of  a  product,  as  to  any 
letter  or  letters,  is  the  sum  of  the  degrees  of  the  highest  terms 
of  the  factors,  as  to  the  same  letter  or  letters  ;  and  so  of  the 
lowest  term. 

E.g.,  in  (rt^  +  2ab  +  b"^)  x{a  +  b)  =  a^  +  ^a'^b  +  3ab^  +  P  the  degree 
of  the  highest  terms  of  the  factors  as  to  a  are  2,  1,  and 
of  the  highest  term  of  the  product,  it  is  3. 

So,  the  degrees  of  the  lowest  terms  as  to  a  are  0,  0,  0. 

6.  If  each  factor  be  homogeneous  as  to  any  letter  or  letters, 
the  product  is  homogeneous  as  to  the  same  letter  or  letters. 
E.g.,  in  the  example  just  above,  the  factors  and  the  product 

are  all  homogeneous  as  to  the  two  letters  a,  b. 

7.  The  whole  number  (f  terms  in  any  product,  before  reduc- 
tion, is  the  continued  product  of  the  number  of  terms  in  the 
several  factors  ;  and  the  product  of  two  or  more  pwlynomials 
call  never  be  reduced  to  less  than  two  terms,  that  of  highest 
degree  a7id  that  of  lowest  degree  as  to  any  letter  or  letters* 


3.  §31  MULTIPLICATION.  45 

QUESTIONS. 

1.  For  how  many  different  values  of  a,  h,  is  the  statement 
(a-bY^a''-)lab  +  b^    true? 

Is  tlie  value  of  (r  —  )lab  +  lr  the  same,  whatever  values 
be  given  to  a,  ^?     Why,  then,  is  the  statement  always  true? 

2.  In  the  product  (^^^  +  2^^^  +  ^^)  x  (/(-[-b)  if  every  b  be  re- 
placed by  an  a,  and  every  a  by  a  b,  is  the  multiplicand  changed  ? 
the  multiplier?  the  product? 

As  to  what  letters  are  these  three  expressions  symmetric  ? 

3.  As  to  what  letters  is  the  product 

(rt^  —  abc  +  ^'j  x{(i-rb  —  c)    symmetric  ? 
Eeplace  a  hy  I,  b  by  3,  c  by  5,  and  test  the  product. 
To  make  the  test  doubly  sure,  replace  a,  b,  c  by  other 
numerals  and  test  it  again. 

Is  the  test  good  whatever  values  be  given  to  a,  b,  c? 

Is  it  a  perfect  test  ? 

What  are  the  most  convenient  values  ? 

4.  Find  tlie  product  (.t  — 2)  x  (2.t  — 1)  and  test  it  by  show- 
ing that  the  sum  of  the  coefficients  of  the  product  is  the 
product  of  the  sums  of  the  coefficients  of  the  factors. 

5.  So,  (x-2)  X  (2;c-l)  x  (x--4)  x  (42' -1)  x  (x-G)  x  (6.Y-1). 
How  many  terms  has  this  product  before  reduction  ?  after 

reduction  ? 

6.  Show  that  the  fourth  principle  is  only  another  way  of 
stating  the  third  when  the  value  1  is  given  to  each  letter. 

7.  In  the  product  a^  +  ^a^-{-3ab^  +  b^  by  a^b,  what  term 
contains  a  with  exponent  0  ?  what  is  the  value  of  a^  ? 

8.  Of  what  degree,  as  to  x,  is  the  product  (x^  +  y)  x  (x^  +  y)  ? 
as  to  y  ?  as  to  :r  and  y  ? 

So,  the  product  (x^y-\-y^)  x  {x^y'^  +  y)  x  (xy^  +  y^)  ? 

9.  Of  what  degree  is  the  product  of 
^m_^^m-i^_^^m-2^2^  .  •  •  +  (( b"' - '' -^ b'"  by  Ti"  +  r^^'^Z*  +  •  •  •  -\- b\ 

Is  this  product  homogeneous  ?  symmetric  ? 

10.  Multiply  4b^-3b^c-3br  +  4c^  by  2b  +  2c,  and  show 
the  application  of  the  seven  laws  just  pointed  out. 


46 


THE  PRIMARY  OPERATIONS  OF  ALGEBRA.   [  II,  Pr. 


ARRANGEMENT. 

Note  2.  Tlie   work   is  often  shortened  by  arranging  the 
terms  of  both  factors,  and  of  the  product,  as  to  the  powers  of 
some  one  letter  and  grouping  together  like  partial  products. 
E.g.,  {a^-{-?>aJ/  +  ?>a'b  +  I/)  x  [b'^-^'Zab  +  a'')  is  written 

a^  +  ^a''b  +  ^ab''  +  b^ 

a^  +  'Zab  ^y 


a^  +  3   ft*6  +  3 

«V+1 

a%^ 

+  2         +6 

+  G 

+  2 

+  1 

+  3 

+  3 

ab" 


+  b^ 


«^  +  5   cH  +  lOa^b^+lOa'b^  +  b    ab^  +  b\ 

CROSS   MULTIPLICATION. 

Note  3.  The  work  is  often  shortened  by  picking  out  and 
adding  mentally  all  like  partial  products,  and  writing  their 
sum  only.     In  ex.  nt.  2  the  computer  says  and  writes 

a^  X  a^  is  a^ 

3a^b  X  a^  is  ^(I'^b,  a^  x  2ab  is  2a^b,  whose  sum  is  5a*b 

3ab^  X  a^  is  3a^%  ^aJ'b  x  2«^  is  Qa^b\  a^  x  b^  is 

aW,  whose  sum  is  lOa^b^ 

h^  X  ft2  is  a'b^  3ab'^  x  2ab  is  Ga^\  3a^b  x  b^  is 

3a^b^,  whose  sum  is  lOa^b^ 

P  X  2ab  is  2ab^,  3ab^  x  b^  is  ^ab*,  whose  sum  is  5«Z>* 

b^  X  b^  is  b' 

and  the  product  is    a^  +  5a*b  + 1  Ou^b"^  +  I0(rb^  +  5ab^  +  b\ 
So,  to  multiply  384  by  287,  product  110208,  384 

i.e.,  3-10^  +  8.10  +  4  by  2.10^  +  8-10  +  7,  287 

the  computer  says  and  writes 

28;  8 

2;  56,  58;  32,90;  0 

9;  21,  30;  64,  94;  8,  102;  2 

10;  24,  34;  16,50;  0 

5;     6,11;  11 

wherein  the  numbers  28,    56,    32,    21,  64,     8,     24,    16,     6, 

aretheprd'ts  4-7,8. 7,4-8,3. 7,8.8,4.2,3.8,8. 2,3-2. 


3,  §3] 


MULTIPLICATION. 


47 


QUESTIONS. 

Multiply,  and  check  the  work: 

1.  (x^  -  2x^ -  32-  +  1)  X  {2x^  -  3x  +  4). 

2.  {l  +  x^-3x-2x^)x{i-{-2x'^-'3x). 

3.  (x^ -hy^  +  z^-\- xy  +  yz- zx)  x  (x -y  +  z), 

4.  {x^-zx  +  y'^  +  yz  +  z^  +  xy)x{z  +  x-y). 

5.  {x^  +  ax-  W)  X  (a;2  +  hx  -  a'-)  x  {x  -  oTb). 

6.  {h'  -  x^  -  ax)  X  (bx  -  «2  +  a-2)  x(a  +  I)-x). 

7.  [(rt-l)2''  +  (r^-l)V+(rt-l)'.r]x[(«  +  l).T+(f;  +  l)3:r-i]. 

8.  (a.-^  —  ax  +  ft^)  X  {a  +  .t)  ;  (ft^  —  x^  —  rr)  x  {x  +  r<). 

9.  (rt-J  +  6-)  X  (b-c  +  a)  X  (c-ft  +  Z/)  x  (a  +  b  +  c). 

10.  (l+.7:  +  a;2  +  r6-3  +  r^-*  +  ri')x(l-2:  +  .T2-a;=^  +  a.-*-a:5). 

11.  (52-5  +  4.r*  +  32^  +  2.6-2  _^  ,^.)  X  (52-^  -  4x'  +  3a;^  -  2x^  +  x). 

12.  (rt*"  +  db'' - 2cP)  x{a-"'-3b-''  +  2c-p). 

13.  (rt'"-26-^  +  3Z/")  X  (3/^-"  +  rt-'"  +  2c-'P). 


14.  x"^  +  <<  I  a;  -\-abxx^-hc 


+  b 


+  d 


X  +  cd',    x^  +  « 


X  +  «^  X  a;^  —  6'  I  a;  +  <?f/. 
-d\ 


15.  2-2 -« 


2  -h  ^^Z>  X  :6'2  +  c  I  2  +  c^/;    x^  —  a\  x  +  ab  xx^  —  c  \x-\-  cd. 


+  d\ 


-d 


1 6.  {x  +  «)  X  [x  -\-b)x  {x  +  c)  X  {x  +  d).  [at  one  operation. 

17.  {x-^  a)  X  {x  +  b)x  {x  -f  c) ;  {x  —  a)x  {x  —  b)x  {x  —  c), 

18.  {x  —  a)x{x-hb)x{x  —  c);  {x  —  a)x(x  —  b)x{x  +  c). 

19.  13x15;  35x79;  234x432;   23.4x4  32;  135.7x12.34. 

20.  18^;     3^;     109^;     163^;     725^;     1881^;     70.23^;    0.2051 

21.  Given     a-hb  +  c=pi,     bc-\-ca  +  ab=p2,     abc=p^; 
show  that     a^  +  W-\-d^=p^  —  2pc,, 

and  that       a'^b  +  ab'^  +  b'^c  +  bc^  +  c^a  +  ra^  =PiP2.  —  ^h  • 

22.  So,  express     a^  +  b'^  +  o^     in  terms  of    PifP^iPz^     using 
the  identity 

a^  +  W^c^-'dabc^{a''  +  b^  +  d'-bc-ca-ab){a  +  b  +  c). 


48  THE  PRIMARY  OPERATIONS  OF  ALGEBRA.      [II,Pr. 

TYPE-FORMS. 

Note  4.  The  work  is  often  shortened  by  the  use  of  certain 
simple  type-forms,  which  the  pnpil  may  prove  by  actual  mul- 
tiplication and  then  memorize.  He  may  also  translate  them 
into  words  and  read  them  as  theorems. 

E.g.,  Form  [2]  may  be  read:  the  product  of  the  sum  and  the 
difference  of  two  numbers  is  the  diff'erence  of  their  squares, 
1]   {x  +  a)x{x  +  b)^x'  +  {a  +  h)x  +  ah. 
E.g.,    (x'  +  6)x(a;-2)  =  a;2  +  (6-2)a;-12  =  ic2  +  4a;-13. 
So,       (a-4)x(«-3)=tJ«-7«^  +  13. 
2]   (a  +  h)x{a-h)^a^-h\ 
E.g.,    (rt  +  5)x(rt-5)=ft2_25. 

So,       {a+J^6)x{a-b^)^a^-h^^  =  a^-W  +  Qh-^. 
3]  (a  +  lf^a'  +  2ah  +  b\ 

E.g.,    {2?/  +  7)^=V  +  2-2^-^  +  '^'  =  4/  +  28?/  +  49. 
So,       (2.^  +  3«)2  =  ix"  +  12ax  +  ^a\ 
4]   {a-hY  =  a^-2ah  +  h\ 
E.g.,    (3;2-8)^=:92^-2-32.8  +  8^  =  92^-48;2;  +  G4. 

So,        (aVb  -cf=a'^2al)  +  W-  2ac  -2bc  +  c\ 

5]   (a  +  ^>  +  c+...)'  =  «'  +  ^'  +  c«+--- 

■^2{ah-\-ac-\- '  "  ^lc-\- '  ""). 

E.g.,    {a  +  h-c-df^a'^  +  W^c^-^-d^ 

+  2(ab-ac-ad-hc-M  +  cd). 

6]  {a-b)  X  (rt"-»H-a"-^Z»-{-a"-«^Ha"-*Z)'-f  •  •  •  +ab''-''-^b''-') 

=  rt"  —  Z*",     if  71  be  any  integer. 
E.g.,    {x-2)x(x^-{-2x  +  ^)=a^-2^  =  x^-8. 
So,       {2c-a^)  X  (86'H4«V  +  2a*c  +  ««)  =  lGc*-«». 
7]  {a  +  b)  X  (««-i-ft«-2^  +  «"-»Z'2-rt"-*Z^3^  . . .  +^Z^"-«-Z>"'0 

—  a'^  —  y^y     if  ?J  be  any  even  integer. 
E.g.,    (.^•  +  2)x(.'c«-2ic*  +  4a^»-8a;HlG.r-32)  =  a.■«-C4. 
8]   (f*  +  ^)  X  (rt«-»-a»-2Z»-|-a»-3d'^-«"-*Z>'+ rt-Z^'-HZ'"-* 

=  «"-|-i",    if  w  be  any  odd  integer. 
E.g.,   (mH3Z)  X  (m'»-3/?i'^^  +  9y/<V^-27m''^/H8lZ*)  =  ?>i'"  +  243Z^ 


3,  §  3]  MULTIPLICATION.  49 

QUESTIONS. 

Name  the  type-form  that  applies,  fiud  the  product,  and  check 
the  work  by  the  principles  of  note  1. 

1.  (a;  +  2)x(a;  +  3).     2.  {x  +  2)x{x-3).    d,  (x-2)x(x  +  ^). 

4:.  (x-2)x{x-d).     b,   (y  +  a)x(y  +  l).    6,  {y -a)x(ij-b). 

1.  {y-a)x{y-Vh).    S,  (y  +  a)x{y-h),    9.  (rt  +  2)(«  +  l). 
10.   {x-6)x(x-Q).  11.   (a-4:)  x{a-3).  12,  (b  +  x)x  (6 -y). 


13.  (x  +  a-\-b)x(x-\-c-\-d),     14.  {x-a  +  b)x(x-c  +  d). 

15.  {3a'x^  +  bb'c')  X  (3a V  +  5 JV). 

16.  (a  +  9)x(«-9).  17.  (3-rt)x(3  +  «).  18.  (2  + a:)  x  (2 -a;). 
19.   (^  +  1)  X  (^•-l).  20.  (x'  +  2)  X  (x'-2).  21.  (a^  +  7)  x  (x^-^), 

22.  (50-5)(50  +  5);     45x55;     78x82;     47x53;     99x101. 

23.  {ax'  +  y')x(ax'-y').  24.  (2a^  +  ^h)  x(2x^-^yh). 
25.  {x-a)x{x  +  a) ;  (af-a^)  x  (x^^  +  «');•••;  (2;"  -  «")  x  (2;"  +  «"). 
2G.  (1  -2:)  X  (1  +  x)  X  (l^x')  X  (1  +2;*)  X  (1  +a:«) .  .  .  {l+x'''). 

27.  Show  that  the  rules  that  apply  to  form  1  apply  also  to 

forms  3,  4.     How  does  the  square  of  the  sum  of  two 
numbers  differ  from  the  square  of  their  difference  ? 
Find  the  values: 

28.  (x  +  Gay.  29.  {3x-2y)K  30.  (a^- 1/2)1  31.  (ab-4:)\ 
32.  {x  +  3yY.  33.  (x-^yf,  34.  (x±^)\  35.  {2x''±3yy, 
36.  (n-hb^)\        37.  (rt-^Tcjl        38.   {^H^±c^)\ 

39.  (100-1)2;    992.    (gi)2.    28^;    73^;     807^;     8.07^;     .08071 

40.  In  the  square  of  a  polynomial  of  five  terms,  how  many  terms 

are  perfect  squares  ?    How  are  the  other  terms  formed  ? 
Find  tlje  values: 

41.  {x  +  y^zf.      42.   {2x  +  3y-4:z)\      43.  {xy  +  yz  +  zxf. 
Find  the  products: 

44.  (a^  +  ax  +  x^)  x(a  —  x)x  (a^  —  ax  +  x^)  x  (a  +  x). 

45.  (a;"-Ha;"-'^  +  2;"-y  +  •  •  •  +xy''-^-\-y''-'')  x  (x-y). 

46.  (2:"-*— a;"-2?/  +  a;"-y ±:r?/"-2H=?/"-^)  x  {x  +  y), 

47.  (j[;+j9r+j[;r^+i?r^+  •  •  •  +^jr"-^)  x  (1  — r). 


50  THE  PRIMARY  OPERATIONS  OP  ALGEBRA.       [11.  tr 

DETACHED    COEFFICIENTS. 

Note  5.  If  both  multiplicand  aud  multiplier  be  such  that, 
when  their  coefficients  are  detached,  the  remaining  factors  of 
pairs  of  successive  terms  have  a  constant  ratio,  the  work  is 
shortened  by  writing  down  the  coefficients  only. 
Take  the  terms  of  loth  factors  in  such  order  that  lohen  the 
coefficients  are  detached  the  parts  left,  taken  two  and 
two  in  order,  have  a  constant  ratio  ; 
in  place  of  the  give^i  polynomials^  write  the  ttvo  groups  of  co- 
efficients, with  0  for  the  coefficient  of  any  alsent  term; 
multij)ly  the  coefficients  as  polynomials;    add  those  partial 
2)roducts  that  pertain  to  like  terms  of  the  final  product; 
in  the  final  product  restore  the  suppressed  letters, 
E. g.,  {a^  +  Za^b -\-ZaW^ W)  x  (aH 2«Z»  +  b^)  [ratio  b  :  a, 

gives    1  +  3+  3+  1 
1  +  2+  1 


1  +  3+  3+  1 

+  2+  6+  6  +  2 
+   1+   3+3+1 

1  +  5  +  10  +  10  +  5  +  1; 
and  the  product  is    a''  +  ^a'b  +  lOa^b""  +  l^^aW  +  hab'  +  b\ 

Check:  1  +  3  +  3  +  1  =  8,        1  +  2  +  1  =  ^, 

8-4  =  32,         1  +  5+10  +  10  +  5  +  1  =  32. 

This  method  is  a  familiar  one  in  arithmetic;  the  ratio  is  10. 

E.g.,    1089x237  =  258093,      or  Ith  +  Oh  +  St-^^u 

237  2h  +  2t  +  lu 

7623  lth  +  Qh  +  2t  +  du 

3267  mh  +  Wi  +  Qh^-lt 

2178  2Jith-\-lfth-^7th-h8h 

258093  2hth  +  5tth  +  Sfh  +  0//  +  9/  +  3  u. 

The  first  form  is  a  case  of  detached  coefficients,  wherein 

the  denominations  and  the  relations  of  the  several  figures  ai*e 

shown  by  their  positions,  as  in  the  last  form  they  are  shown 

by  words  and  signs. 


3,  §3J  MULTIPLICATION.  51 

QUESTIONS. 

Multiply  and  check  the  work : 

1.  (2x  +  3)x(3x-4).         2.  (.^2  +  3.i'  +  2)x(.T2-3a;  +  2). 

3.  {3y  -  5)  X  {2y  +  7)  x  (2  -  4/)  x  (1  +  2f). 

4.  {x^  +  3x^y  +  3xf  +  y')  x  (.r^  +  2rr //  +  t/^)  x  (.-i^  +  ?/). 

5.  (2.r^  -  ^x^y  +  2.?/^)  x  (2.c=^  +  3xy^  +  2y^). 

6.  (2^-5)^.       7.  (y  +  2y^  +  3y')\        8.  (2-32-3^^2^^)2. 
9.  (r^  -  3:^2?/^  +  3^«/*  -  V^)  x  (^*  -  ^^^""f  +  6.^^*  -  4a;^^  +  tf). 

10.  (.^3-2a;2^i)x(2:«2-32;  +  4)x(x  +  l). 

11.  {x^—mx  +  ?;i^)  X  (.?;2  +  mx  +  ??i^)  x  (x^  +  ??iV  +  ?m*). 

12.  {caf  +  hyh^-cyh^)  x  {ayh'^-hyh^  +  cyz^). 
Show  that 

13.  a;x(a;  +  l)  x  (a;  +  2)  x  (a:  +  3) +1  =  (a;H  3.^  +  1)1 

14.  (.V-l)xyx0/  +  l)x0y  +  2)  +  l  =  0y'  +  ?/-l)l 

Show  that  if  y  be  replaced  by  a;4-l,  the  equation  in  ex.  14 
becomes  the  equation  in  ex.  13. 

15.  By  successive  multiplications,  and  preferably  by  detached 
coefficients,  find  the  first  five  powers  of     a^-h     and  of     a  —  b» 

16.  So,  of    x-\-y    and  of    x  —  y,    of     h  +  h     and  of    U  —  k, 

17.  From  observation,  and  comparison  of  the  results  above, 
state  a  general  principle  that  holds  good  as  to: 

1.  The  number  of  terms  in  any  power  of  a  binomial. 

2.  The  exponents  of  the  first  letter  {a,  x,  or  h)  in  the 
successive  terms  of  any  power. 

3.  The  exponents  of  the  last  letter  {b,  y,  or  k). 

4.  The  signs  of  the  terms. 

5.  Tiie  coefiicients  of  the  first  term,  the  second,  and  last. 

18.  Show  that  the  terms  of  any  power  of  a-\-d  are  homo- 
geneous, and  that  terms  equidistant  from  the  ends  have  the 
same,  or  opposite,  coefficients. 

AVhat  terms  of  powers  of  a-\-b  are  identical  with  terms 
of  like  powers  of     a  —  b,    and  what  are  opposites  ? 


52     THE  PRIMARY  OPERATIONS  OF  ALGEBRA.   [H.Pr. 

SYMMETRY. 

Note  6.  The  work  is  often  shortened  by  noting  any  sym- 
metry that  may  exist  among  the  factors,  singly  or  in  groups. 
E.g.,  the  product  (2a  +  b  +  c)  x  (a  +  2b  +  c)  x{a-{-b-\-2c),  has 
the   terms   2a'a-a  =  2a^, 

2a-a'b  +  2a-2b'a  +  b-a-a  —  la^b, 
2a'2b'2c  +  b'C'a  +  C'a'b-\-2a'C'b  +  b'a'2c-hc-2b'a 
=  16abc; 
and  '.•  every  term  of  the  product,  being  entire  and  of  the 

third  degree,  is  of  like  form  to  one  of  these, 
and      the  product  is  symmetric  as  to  a,  b,  c; 
.*.  it  has  the  terms  2b^,  2(^  as  well  as  2ct^, 
and       Wa,  ric'a,  lab\  lb(^,  lca\  as  well  as  la% 
and  the  whole  product  is 

2^3  _^  2b^  +  2^.3  +  7«2^  +  Wc  +  Ic'a  +  laW  +  Uc^  +  Ica^  +  IQabc. 

So,  in  the  product  {2a-\-b-c)x{2b  +  c-a)x  {2c  +  a-b): 
the  terms  in  a^,  P,  c^  have  the  same  coefficient,  —2, 
those  in  a^,  b^c,  (?a     have  the  same  coefficient,      5, 
those  in  aW,  bc^,  ca^     have  the  same  coefficient,  —1, 
that  in  abc  has  the  coefficient  2; 

and  the  product  is 

-2{a^-\-b^  +  6-3)  +  5  {(t'b  +  b'^c  +  chi)  -  (ab^  +  bd'  +  ca^)  +  2abc, 

So,  to  find  the  sum  (a^b-2cY-\-{b^-c-2af^-{c  +  a-2bY'. 

by  multiplication,  or  from  the  type-form  {a  +  b^ )^,  get 

(a  +  b-  2cY  =  a^  +  Z/H  ^c"  +  2ab  -  Uc  -  4:ca, 

by  symmetry,  write, 

(h  +  c-2aY  =  b^-^c^  +  4a^  +  2bc-4ca-4ab, 
(c-]-a-2bY=c'  +  ci'  +  ib^  +  2ca-iab-4bc, 

and  add  ;  the  sum  is     C  {a^  -}-b^  +  c^-  be  -  ca  -  ab). 

In  such  symmetric  expressions,  where  three  or  more  letters 
are  involved,  these  letters  may  be  kept  advancing  in  cyclic 
order,  abc,  bca,  cab,  ab,  be,  ca,  as  if  they  were  points  on 
a  circle  following  one  another  in  the  same  rotary  direction. 


3,53]  MULTIPLICATION  53 

QUESTIOXS. 

Multiply,  add,  and  check  the  work: 
1.  {x  +  yY  +  {x-y)\  2.  (x  +  yf  -  (:c  -  y)\ 

3.   {x^y  +  zy  +  (:c-y  +  zY.  4.  {x  +  y  +  zY-{x-y +  z)\ 

5.  (x  +  y-zy  +  {x-y  +  z)\  6.  (x  +  y -zy-{x-y  +  z)\ 

7.  (  —  x-\-y-\-z)x(x-y  +  z)x(x-\-y  —  z). 

8.  {-a  +  b-^c  +  d)-{a-h  +  c  +  d)-{a  +  b-c  +  cJ)'{a  +  l-\-c-d), 

9.  {ax  +  ^«/) •  {bx  +  rt^)  +  {ax  —  by)-  {bx  —  ay). 

10.  {a  +  b)-{c  +  d)^-{a  +  b)'{c-d)  +  {a-b)'{c  +  d) 

+  {a-b)'{c-d). 

11.  (r/.  +  ^  +  c)-(2;  +  ?/  +  2)  +  (r^  +  ^-r)-(./-  +  ?/-2) 

+  {a-b  +  c) '  {x-y  +  z)  ^  {- a  +  b  +  c)  '  {-x  +  y  +  z). 

12.  (Z'c  +  rtf/)2  +  {ca  +  ^^f/)2  +  {ab  +  6y/)1 

If    25  =  rt  +  &  +  c,     prove  the  identities  below,  and  check  the 
work  by  putting     a  —  b  —  c—l. 

13.  a^-{b-cf^^{s-b){8-c).  14.   {b  ^  cf  -  a^  ^  \s{s  -  a). 
15.  {s-aY^{s-bY^{8-cf^H^:=^a^^b''^c', 

16.     (.S-«)3+(.s_J)3+(,._c)3_^.3  33_3^^J^, 

17.  %{8-c{){8-b){s-c)-Va{s-b){s-c)^b{s-c){s-a) 

+  C(5  —  «)  (5  —  ^)  —  «^C. 

18.  166-(.s -a){8-  b) {8 - c)  =  2(Z/V  +  cV 4- r^^^')  -  («*  +  Z>*  +  c*). 

19.  lix  +  y  +  z  —  Oy    show  that    oi?  +  y^-{-^  =  ^xyz',    hence  thai 

{b-cf  +  {c-af+{a-bY='d{b-c)'{c-a)'{a-b). 

20.  {x^  4-  2-2)  •  (1/2  -  ^i)  +  (.^2  +  x^) .  (^3  -  y^)  +  (a:3  +  2:1)  •  (y^  -  y,) 

=  ^12/2  -  2:2^1  +  2:2?/3  -  Xsy2  +  .Tg^/i  -  0:1^3. 

2 1.  (^f a:  +  by  +  cz)  •  {bx  +  c?/  -\-az)-  {ex  -\-  ay  +  bz) 

=  abc{x^  +  y^  +  z^)  +  {a^  +  b^  +  c-^)^:,^/^  +  3abc  •  a;?/;?; 

+  (rt2^>  +  b^c  +  c-^ft)  •  {r'y  +  ?y  ^2  +  z^a:) . 

22.  {ax  +  %  +  czY  +  («2:  +  c?/  +  ^2)^  +  {bx  +  rtz/  +  cz)"^ 

+  ( Ja;  +  c?/  +  «2)2  +  {ex  -\-ay  +  bzY  +  {ex  -\-by  +  azY 

=  2{a^  +  b''  +  c'){x''  +  y^  ^z^)  +  4:{bc  +  ca  +  ab){yz  +  zx  +  xy). 


54      THE  PRIMARY  OPERATIONS  OF  ALGEBRA.   [II,  PR. 
CONTRACTION. 

Note  7.  When  only  the  first  few  terms  of  a  product  are 
wanted,  the  work  is  shortened  by  omitting  all  partial  products 
that  do  not  enter  into  the  required  terms. 

E.g.,  to  find  {l-3x  +  5x^ )^  as  far  as  the  a^-teim: 

1-3:^  +  5.^2 oj.         1-3  +  5 • 

l-3a;  +  5a;2 1-3  +  5 


l-^x-^-bx" 

1-3  +  5 

-3x  +  9x^ 

-3  +  9 

+  5x^ 

+  5 

1-Qx-hl^x^  1-6  +  19 

So,  omitting  x*'  and  higher  powers,  to  find  the  product, 

(l+a-+a;2+. .  .)x{l-2x+3x' )x{l+4:X+9x^-\-"  •)  : 


write     1111x1  -23  -4  I'l     2-2x1     4     9     16 

-2  -2  -2  4-4     8 

3     3  9-9 

-4  16 

1-1     2-2  1     3     7  13 

and  the  product  sought  is  l  +  3.^'  + W  +  13a.-^. 

This  method  of  contracted  multiplication  may  be  used, 
with  great  profit,  with  decimal  fractions. 
E.g.,  to  find  the  product  37.8562x14.9716,  correct  to  two 
places,  and  .2819  x  .3781  x  .2148  to  three  places. 

37.8562        and         .2819  .1065 

14.9716  .3781  .2148 


378.562 

.0846 

.0213 

151.425 

197 

11 

34.070 

23 

4 

2.650 

.1065 

1 

38 

.023 

23 

566.77 

In  writing  down  the  i)artial  products,  carry  what  Mould 
have  been  carried  had  the  multi])lication  been  made  in  full. 
E.g ,  the  partial  product  23  =  3-6  +  5  carried  from  78 •  6. 


3,  §3]  MULTIPLICATION.  55 

QUESTIONS. 

Find  the  products  (or  powers)  as  far  as  the  a;*-term: 

1.  (l-\-x^x'+-")'{l-x  +  x'' ). 

2.  {l-]-2x  +  3x^+  .  •  ')'{l-2x-\-^x^ ). 

3.  {l-\-^x-\-bT^)'{l  +  3x)-(l  +  bx'). 

4.  {Z-5x-{-9x^)'(l-6x)'{l  +  9x^). 

5.  (l-2x  +  ^x''-4:X^-\-6xy.            6.   (1  -  ix  +  p^  -  ^:^ -\- ix% 
7.  {?>  +  5x-h'7x'+'"y.  8.  (i-ix  +  \:^ f. 

9.  {a-{-bx-]-cx''i-  "  -)'(a'  +  yx-\-c'x'^+  "  '). 

10.  {a-bx  +  cx"^ )-{a'-yxi-c'x^ ). 

11.  {a-\-I)x  +  c:^-\-dx^+"-Y.        12.  (a-bx-\-c:iy^-dx^-^  -  "f, 
13.  {a  +  bx  +  cx'  +  d:t^+'-'Y.        14.  (a-bx-\-cx^-dx^+ -- -f, 

Fiud  the  products  (or  powers)  as  far  as  the  a;^-term,  with 
three-figure  decimals: 

15.  {l-\-.ox-\-,(}9x')'(l-.bx  +  .09x^). 

16.  (3  +  .ox-.0:x')'{d-.6x-h.07x^). 

17.  (1 -.07^2) -(3 +  .0092^2).  18.   (l  +  .007r').(3-.09a;2). 

19.  {i-^.ie7x+,oux^+.ooix^y. 

20.  (l-.167x  +  .014a;2-.00U-^)l 

21.  (1 +  .056^2 -.006:^3)1  22.  (l-.3332:  +  .006a;2)l 

23.  (3  + . 5a;  +  .07 x^  +  .009^;^)  •  (5a;  +  .070;^)  •  (7  +  .0090;^). 

24.  (1  +  .07a:) .  (1 4-  .07a;2)  •  (1  +  .07a:^). 

25.  (l  +  .07a;7.  26.   (l+.07a:^)^  27.   (l  +  .07a;*)*. 
Find  the  values  correct  to  thousandths,  when  a;  =  .l: 

28.  (l  +  2a:)^        29.  (l  +  2a;  +  3a;2)l        30.  {l  +  2x-\-3x'  +  ^a^)\ 
31.  (a;  +  5)^  32.  (x^'-x  +  by.  33.  (^^-x-bf. 

Find  the  values  correct  to  thousandths  when  a;  =  .02: 

34.  (l  +  2a:)^        35.  (1  +  2a;  +  3a;2)^        36.  (l  +  2a;  +  3a;2  +  4a;^)3. 

37.  (a: +  5)*.  38.  (x'-x-[-6y.  39.  (x^-x-by. 

40.  li  x  —  a-\-by  -{- cif-  +  dif  +  •  •  • ,    y  —  l  +  mz  +  nz^ -\-p^  +  •  •  •> 
find  the  value  of  x  in  terms  of  z  as  far  as  the  2;^-term. 


56  THE  PRIMARY   OPERATIONS  OF  ALGEBRA.       [U.Pr. 

§4.   DIVISION. 

PrOB.  4.   To    DIVIDE   ON^E   NUMBER   BY   ANOTHER. 

{a)  A  monomial  hij  a  monomial:  to  the  quotient  of  the  mime- 
rical  coefficients^  annex  the  several  literal  factor s,  each 
tahen  as  many  times  as  the  excess  of  the  exponeiit  of  the 
dividend  over  that  of  the  divisor;  [I  th.  10  cr. 

marh  the  quotie7it  positive  if  the  elements  he  both  positive  or 
hoth  negative^  and  negative  if  one  element  he  p>ositive 
and  the  other  negative.  [inv.  pr.  3. 

E.g.,     ^Za-^hl^  :  7a(fd'=9a-'b'c-\ 

-35x*7/-V  :  5.r?/-V=  -"Tx^y-h-^ 

i^a-^-M-'  :  -^ac-hl-'=  -|«-^Z»-V, ' 
-Ix-yz-^  :  ~ix-'y-h^=::^Sxy'z-K 

(h)  A  polynomial  by  a  monomial :  divide  each  term  of  the  divi- 
dend by  the  divisor;  add  the  partial  quotients.   [I  th.  7. 

E.g.,  (45;r^/2  + 105.-^2-1 -165a;-2^-^;2-*)  :  -IhxyH-'' 
=  ~^xh^-ly-h  +  llx-hj-^z-\ 

{c)  A  p)olynomial  by  a  polynomial :  arrange  the  terms  of  boJi 
polynomials  as  to  the  powers  of  some  one  letter; 

divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor;  multii^ly  the  whole  divisor  by  this  partial 
quotient,  and  subtract  the  product  from  the  dividend; 

repeat  the  ivork  upon  the  ronainder  as  a  new  dividend; 

add  the  partial  quotients;  their  sum  is  the  quotient  sought, 
and  the  part  of  the  dividend  left  undivided  is  tlie 
remainder,  [I  th.  7  cr. 

E.g.,  to  divide  a^  +  b'^hy  a-^-hi 


write    a^-\-b^ 
a^^-a^b 


a  +  b 
a^-ab  +  b^ 


-a^-i-b^ 
—  a^b  —  ab 


ab^  +  b^ 
ab^  +  b^ 


4.  §4] 


DIVISION. 


67 


QUESTIONS. 

Find  the  quotients  below,  and  check  the  work: 


1.  3a'b:ak 


2.  -3ax:  -x\ 


3.   77171-^:  -7)1^11, 


4. 

7. 

9. 
11. 
12. 
14. 
16. 
18. 
19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 
28. 
29. 
30. 
31. 
32. 


2r-W 


5.  51a'b'c:-l'7a^.    6.  23U''^  +  y :  3.t> 
x''  +  2ax-^b):x.  8.   {ix^-ixy-''-\-^y-'):  -dx'y-K 

,i^-x-12):(x-i).  10.   {4:x'-12x''  +  9):(2x'-3), 

15x^  +  x^y-^  +  iij-^):(3x  +  2y-'). 
a^v'^ -  y') :  (ax"" - y^).  13.    (x^  - 2x^y  -  3?/) :  {x"  +  y). 

^2n  _  ^2n)  .  (^,n  ^  ^ny  ^D,    (9«V  -  496'y )  I  {3aV  _  r^^^^y 


l-x'):{l+x). 


17.  (c«-3c*  +  3c2-l):(c2-l). 


«r"-rt):  (r-1);     («-«r"):  (1-r). 

a^-2ab-2ac-\-b^-^2bc-\-c^):(a-b-c). 

a'-b'  +  2bc-(f):(a  +  b-c);   (b^  +  2bc  +  (f-a'):{b  +  c-a). 

a^  —  7n\v^  +  277inx^  —  7i^x^) :  {a  —  7nx  +  717?), 

cv>-x')'.{a-x)',     (a'-y^):{a-yy,     {x'-y'):{x'-y'). 

a'-\-a^):{a-\-x);    {a^ - y^) -.  [a  +  y) \     (x^  +  y^) :  {x""  +  y% 

a;«+H3.^-42;"-2  +  6.r):(a;2  +  2). 

x^  +  ax -r  bx  +  CX  +  clx -\-ac  +  acl-[-bc-{- bd)  :(x  +  a-\-b). 

I2m\i?  -  llm^x^  +  lOmx -  3) :  (hnx  -  3). 

ft»^  +  2_3^n  +  l_^n_^2ff«-^-4«'^-2):(a2-«  +  l). 

2/  +  3?/-7  +  15y-^-8?/-^4-52/-^):  (2-^-1  +  2^-2). 
x^-22a^-\-60a^-6Dx'^-\-12x-{-4:):(x^-\-6x  +  l). 
x^y  +  .r^2  +  x^z  +  2a:?/2;  +  ^z^;?  +  xz^  +  t^^^)  :  (y  +  2;)  :  (a:  +  y). 
x^  -  4:7^y  +  6.^2^2  _  4^^3  _f.  ^4j .  (^  _^  2a;?/  +  y^) 
:  (x^-2a^y  +  2xy^-y^). 


.  x^  +  a 

x^  +  ab 

x^'  +  abc 

+  b 

+  ac 

^abd 

+  c 

-had 

-\-acd 

^d 

+  bc 
-hbd 
-\-cd 

+  bcd 

X  +  abed 


+  b 


x  +  ab 


58     THE  PRIMARY  OPERATIONS  OF  ALGEBRA.   [n,  Pr- 

CHECKS. 
The  work  is  tested  by  multiplication  and  sometimes  by  tbe 
principles  of  prob.  3,  note  1. 

FORM  OF   QUOTIENT. 

Note  1.  With  some  changes  that  are  apparent,  the  princi- 
ples of  prob.  3,  note  1  hold  true  in  division. 

ARRANGEMENT. 

Note  2.  Unless  the  division  be  exact,  the  quotient  and 
remainder  are  different  for  every  different  choice  of  trial 
divisor ;  but  the  complete  quotient,  including  the  fraction,  is 
always  the  same. 

E.g.,  (ic^  +  1)  :  (x  +  1)  gives  for  quotient  and  remainder: 
x  —  lf     2    when  a;  is  trial  divisor, 
1  —  x,     2af    when  1  is  trial  divisor. 
So,  {x^ -\- y^ -\- z^) :  {x-{-y  +  z)  gives  for  quotient  and  remainder: 
x-y-z,    2{y\-\-yz  +  z^)     when  x  is  trial  divisor, 
y  —  z  —  x,     2{z^ ■\- zx  +  a^)     when  y  is  trial  divisor, 
z  —  x—y,    2{x^+xy  +  y^)     when  2  is  trial  divisor, 

DETACHED   COEFFICIENTS. 

Note  3.  If  both  dividend  and  divisor  be  such  that,  when 
their  coefficients  are  detached,  the  remaining  factors  of  pairs 
of  successive  terms  have  a  constant  ratio,  the  work  is  shortened 
by  writing  down  these  coefficients. 

Li  place  of  the  given  j^olynomialSf  write  the  tioo  gro2ips  of  co- 
efficients, with  0  for  the  coefficient  of  any  absent  term; 
divide  these  coefficients  as  in  the  division  of  polyiiomials,  and 

restore  the  stipj^ressed  letters. 
E.g.,  {a^  +  h^)  :  {a  +  b)  gives 


10     0     1 

1     1 

1   1 

1-11 

-10     1 

-1  -1 

1   1 

and  the  quotient  i 

ia^-ah  +  b\ 

4,  §4]  DIVISION.  69 

QUESTIONS. 

1.  Show  that  the  rules  for  the  coefficients,  exponents,  and 
signs  of  a  quotient  are  a  direct  consequence  of  division's  being 
the  inverse  of  multiplication. 

2.  State  the  first  six  of  the  principles  in  prob.  3,  note  1,  in 
such  form  that  they  will  apply  to  division. 

3.  Show,  as  a  consequence  of  the  seventh  principle,  that  a 
monomial  is  not  divisible  by  a  polynomial,  and  that  when 
there  is  no  remainder,  the  first  and  last  terms  of  the  dividend 
are  divisible  by  the  first  and  last  terms  of  the  divisor. 

4.  Find  the  quotient  {l+co^-8f  +  6xi/):{l-\-x-2i/),  hav- 
ing it  arranged  in  the  order  of  rising  powers  of  x^  and  the 
coefficients  to  falling  powers  of  i/. 

5.  So,  (18xijz  +  27z^-3^  +  8i/^):{x-3z-2y). 

6.  By  detached  coefficients,  divide  x^  + 10.?;^  +  dox^  +  dOx  +  24 
in  turn  by    xi-1,  x  +  2,  x  +  3,  x+4:,     and  check  the  work. 

7.  So,     a;*-5a;2  +  4     by    x-2,    x-1,    x  +  1,    x-{-2, 

8.  So,     x^-10x^  +  3ox^-bOx  +  24:hyx-l,x-2,x-3,x~4:. 

9.  So,     x^  — 5x^1/^ ■{■4:2/*    by    x~2y,    x  —  y,    x  +  y,    x  +  2y, 

10.  So,     x*-10x^y  +  doafy'^-50xy^-{-24:f   by   x-y,  x-2y, 

x-'^y,x-4y, 

11.  So,     ii^  +  42;3-6:c2^7a;-2     by    x?^^x-2,     a^-x  +  1, 

12.  So,     a'  +  ^a'b - Da'b''  +  6aP-b'hj a^-hoaI)-b%  a^-ai  +  bK 

13.  Divide  (i^-2x*  +  3x^^4:X^-\-DX-Q  by  x-2,  the  quo- 
tient by  x  —  2,  and  so  on.  Write  the  last  quotient  and  the 
numerical  remainders  in  their  order,  as  coefficients  of  powers 
of  a;-2. 

14.  So,  a^-\-2x*  +  33^-\-4:X^+5x-\-6  by  x-\-2,  the  quotient 
by    x  +  2,    and  so  on. 

15.  So,  x^  +  2x'y  +  S:^y^  +  4:X^y^  +  5xy*^h6f  by  x  +  2y,  the 
quotient  by    x  +  2y,    and  so  on. 

16.  Compare  the  work  in  exs.  13-15  with  that  of  reducing 
seconds  to  minutes,  hours,  and  days  by  the  ordinary  process  of 
reduction  ascending:  what  is  the  scale  in  ex.  14  ?  in  ex.  15  ? 


60  THE   PRIMARY   OPERATIONS  OF  ALGEBRA.       [n,PR. 

TYPE-FORMS. 

Note  4.  If  the  dividend  and  divisor  can  be  reduced  to  known 
type-forms  [pr.  3,  nt.  4]  the  quotient  may  be  written  directly. 

E.g.,  {x'-^7x  +  U)  :  {x  +  4)  =  xi-d,  [1 

i^-y')'(^-^y)=^-y.  [2 

{^  -  /) :  (^  -  y') = ^' + ^y + ^Y + /•  l^ 

SYMMETRY. 

Note  5.  If  both  the  dividend  and  divisor  be  symmetric  as 
to  two  or  more  letters,  and  if  there  be  no  remainder,  then  the 
quotient  will  be  a  symmetric  function  of  the  same  letters.  It 
is  then  often  sufficient  to  get  a  few  characteristic  terms,  and 
to  write  the  rest  therefrom  by  symmetry. 
E.g.,  {x^y  +  xif  +  yh  +  yz^-{-z\v  +  zx^  +  3xi/z)  :  {x-\-y  +  z), 
gives  xy  for  one  term  of  the  quotient; 

.*.  yz  and  zx  are  also  necessary  terms  of  the  quotient; 
and*.*  the  product  (x-l-y-{-z)'{xy-\-yz-\-zx)  is  the  dividend, 

.*.  xy  +  yz  +  zx  is  the  quotient  sought. 
The  pupil  must,  however,  use  great  caution  if  he  employ 
symmetry  in  division;  he  may  use  it  as  suggestive  of  the  true 
answer,  but  hardly  ever  as  conclusive,  because  he  cannot  be 
sure  beforehand  that  there  is  no  remainder. 

CONTRACTION". 

Note  6.  When  only  the  first  few  terms  of  a  quotient  are 
wanted,  the  work  is  shortened  by  omitting  all  partial  products 
that  do  not  affect  the  required  terms. 
E.g.,  (I -\- X -\- x^  +  x^)  :  (l  —  2x  +  3x^  —  4x^)  to  four  terms  gives 


1+  x-^  a^i-  x^ 
l-2x-\-^x'-43^ 


l-2x  +  3x^-4Lay'        1111 
l-f3a;  +  4ar'  +  4a;3        1-2    3-4 


1-2    3-4 


13    4    4 


dx-2x'-\-6a^  3-2    5 

dx-6x^-^9a^  3-6    9 

4^^=4^  4^ 

42;2-8r'  4  "8 

4?  4 


4,  §4]  DIVISION.  61 

QUESTIONS. 

Name  the  type-forms,  find  the  quotients,  and  check  the  work: 

1.  (x'-2ix  +  10i):{x-U).  2.   (x'-y*):{x^-if), 

3.  {x---^x-12):(x-6).  4.   (io0-\-'3lxij i-xY):  {5+xy). 

5.  (2  +  X - x^) :  (2  - x),  6.   (2.7;^ -x- 15)  :  (x - 3). 

7.  {4x'  -  9y') :  (2^  +  3f).  8.   {a^^  -  ^9x')  :  (ab  -  7x% 

9.  (x2'^-/"):(a;"±i/").  10.  {l-lOOa'b'v'):  (l-\-10a^bc^). 

11.  [l-{7a-3bY]:(l+7a^3b). 

12.  [(a^  +  a')'{x^-a')]:[{x'  +  ax  +  a'')-{a^-ax  +  a')]. 

13.  [{a  +  by-c^y.ia  +  b-c).  14.  [a^-{b-cY]:  (a-b  +  c). 
15.  (a2  +  2aa;2  +  a;*);(a  +  a;2).  16.  (x^-8x-hlQ):{x-^). 

17.  (4.?;«  -  12.T^  +  9/) :  (2a;  -  3/y). 

18.  (49a2  +  42rtZ>  +  9Z/^):(7«  +  3^), 

19.  (4^2  +  J2  4-  c2  _  4rt J  +  4rtc  -  2Z>c) :  (2a  -b  +  c). 

20.  (a^^^ - 343:1'^)  :(rt^- 7a;).            21.  (a^  +  2l6P):{a  +  Qb). 
22.  [(.r  +  ?/)H2^:(2;  +  ^  +  ;2).          23.  [.r^ -  ( ?/ -  2)3]  :  (:c  - ,y  +  ^). 
24.  (a'''-b'''):(a  +  b).                     25.  (a2"+H^^'«+i) :  («  +  ^>). 
26.  (a;"*»-l) :(.'£"» -1).                     27.  (a;"*"-!) :  (.^'"-l). 

By  symmetry,  find  the  quotients  below,  and  check  the  work: 

28.  [3abc  +  a"(b  +  c)+b%c  +  a)+c%a  +  b)]:{a  +  b  +  c). 

29.  (a'  +  P  +  (^-3abc):(a  +  b  +  c). 
30„  (a^-b^  +  (^  +  3abc):{a-bi-c), 

31.  («8  +  a'b'  +  «*<?/*  +  a^'  +  b') :  (a*  +  a^  +  a^^^  +  ab'  +  ¥). 

32.  (a*62  -  a'-b'  +  Z/*c;^  -  b^c'  +  cV  -  cV) 

;  {a'b - ab^  +  Vc -b&^- &a - ca\ 

33.  \x\y~£)^y\z-x)^z^(x-y)\ 

,l:c\y-z)^y\z-x)^z\x-y)\ 
Find  the  quotients  to  the  :r*-term,  using  three-place  decimals  : 

34.  (1-.2.C  +  . 042-2 ):(l  +  .U-  +  .01a;2+--0- 

35.  (1  4-. 2:c  +  . 04:^:2^  .  .  .):  (l-.l^c  +  .Ola;^ ). 

36.  {x-\:^^^\^x^ ) :  {\-\x}^i-^x^ ). 


62 


THE  PRIMARY  OPERATIONS  OF  ALGEBRA.   [II.  Pr. 


SYNTHETIC    DIVISIOif. 

Note  T.  If  the  coefficient  of  the  first  term  of  the  divisor  be 
1,  the  work  of  division  by  detached  coefficients  is  shortened 
by  the  method  of  synthetic  division.  It  is  a  species  of  short 
division,  in  wliich  the  same  numbers  play  the  part  of  both 
remainders  and  quotients. 
E.g.,  to  divide  x^  +  bx^rSx  +  4.  by  a;  +  2: 
write  15     8     4 

2         2     6     4 
1    1     3     2,    0 

The  divisor  stands  at  the  left  in  a  vertical  column,  reading 
upward;  the  part  products  2,  6,  4  are  directly  below  the  like 
terms  of  the  dividend;  of  the  figures  1,  3,  2,  0,  in  the  lower 
line,  1  is  the  first  quotient  figure,  3  is  first  a  remainder  stand- 
ing for  32;^,  then  a  quotient  figure  standing  for  3rr,  2  is  first  a 
remainder  standing  for  2.r,  then  a  quotient  figure,  and  0  is 
the  final  remainder.     The  quotient  is  x^  +  2>x-\-2. 

It  is  customary  to  change  the  signs  of  the  terms  of  the 
divisor  after  the  first,  and  so  of  the  resulting  part  products; 
the  subtractions  are  then  done  by  addition,  and  the  work 
takes  the  form 

15     8     4 
-2  -G  -4 


"2 
1 


1     3     2,    0 
So,  {x^  +  bx^  +  ^x  +  Q>)  :  (a;2  +  3a;-f  2)  gives 
1     5     9     G 
-2  -4 
-3  -6 


-2 
-3 

1 


1     2,    1     2 

wherein  both  3  and  2  liave  tlieir  signs  changed,  and  their 
products  by  1,  the  first  quotient  figure,  give  the  "3,-2  in  the 
body  of  the  work;  tlie  sum  5+^3  is  2,  the  first  remainder 
and  tlie  second  quotient  figure;  and  "G,  "4  are  the  part 
products  of  -3,  "2  by  this  2. 

The  quotient  is  x-\-2,  and  the  final  remainder  is  a; +  2. 


4,  §41  DIVISION.  63 

QUESTION'S, 

1.  Show  that,  with  a  divisor  in  which  the  coefficient  of  the 
first  term  is  unity,  each  term  of  the  quotient  has  the  same 
coefficient  us  the  first  term  of  the  dividend  used:  i.e.,  that  the 
same  number  plays  a  double  part,  being  the  coefficient  of  the 
first  term  of  the  remainder  and  of  the  new  term  in  the 
quotient,  and  that  it  need,  therefore,  be  written  but  once. 

2.  Show  that  changing  the  sign  of  the  divisor  and  adding 
the  partial  products  to  the  dividend  does  not  cliange  the  value 
of  the  remainder,  and  therefore  not  that  of  the  quotient. 

By  synthetic  division,  find  the  quotients,  and  check  the  work: 

3.  (r^- 5^:2 -46^ -40):  (.^  +  4).  4.  (.^^-40)  :(.-«  + 4). 

5.  {x^  +  x^-bx'-llx-^0)i{x-d),  6.  (a;*-5.i-2-30):(:c-3). 

7.  (32;^-5a;2-7.T  +  10):(a;-2).  S,  {'ix^-lx  +  10)\{x'-2). 

9.  (4a.-5  +  17a;*  +  9a;3-20.t2-3a;  +  9):(.'e  +  3). 

10.  (x'-l)\{x-l),  11.  {x^-y^):{x^y). 

12.  {x^-y^)\{x  +  yy  13.  {:x?-x^y^xy^-y^):{x-yY 

14.  {Q?-^x^y  +  ^xy^-^f)x{x-y).  15.  (6.r*- 96) :(:?;- 2). 

16.  (a^-aW-^'ZaW-Uy^ia^-ah  +  W),      17.  (a^-h'):(a^  +  ¥), 

18.  (:c«  +  10a;-33):(ie2-2a;  +  3).       19.  {x^  +  lOx-Z^):^^?^-^). 

20.  {Zx^-4.x^-x^  +  2Zx^-2Sx  +  lb):{x'-2x  +  ^y 

21.  (a*  +  a^b  -  Sa%^  +  l^ah^ -  lb¥) :  {a^  +  3ab  -  5b^). 

22.  (4:z'-18z^-lQz'-78z  +  54:):(^-2z^  +  z-9), 

23.  (4:X^-6:(f-\-8x^-10x^-8x^-5x-4:):{x^~2x^  +  3x-4). 

24.  (a;Hl51ic-264):(a;2-4.T  +  ll).  25.  {x^-264):(x^  +  ll). 
26.  {2u^-82x~2^0):{x''  +  4:X  +  b),  27.  (2:c^-240) :  (a;H5). 
28.  {x^-\-x^-Ax'  +  5x-d):(x'  +  2x-3).       29.  (x* - 3) :  {x^ - 3). 

30.  (x^-22x'  +  60x^-55x^  +  12x  +  4:):(x^-3x  +  2). 

31.  Divide     3x^-5x^^7x^-9x  +  n     hy     x-3,    the  quotient 

by     x-3,     and  so  on;  write  the  last  quotient  and  the 
remainders  as  coefficients  of  powers  of     (x  —  3). 

32.  So  divide   3x^-\-b.i^-i-7x''  +  9x  +  n  by  a:  +  3,   by  x  +  3,-" 


64  THE  PRIMARY  OPERATIONS  OF  ALGEBRA.     [II.Prs. 

§5.    FRACTIONS. 
PrOB.  5.    To  REDUCE  A  SIMPLE  FRACTION  TO  LOWER  TERMS. 

Divide  loth  terms  hy  any  entire  numher  that  divides  them  with- 
out remainder;  the  quotients  are  the  terms  of  the  re- 
duced fraction,  [I  th.  2  cr.  4. 

E.g.,  'iQa'b^cy24:a''bx=Ucy2ax.  [div.  12  a'b. 

For  reduction  of  fractions  to  their  lowest  terms,  see  IV, pr.  4. 

PrOB.    6.     To    REDUCE   A    SIMPLE    FRACTION"    TO    AN    EQUAL 
FRACTION   HAVING   A   GIVEN   NUMERATOR  OR   DENOMINATOR. 

Divide  the  given  new  numerator  or  denominator  ly  the  old 
one,  and  multiply  both  terms  of  the  fraction  by  the 
quotient,  [I  th.  2  cr.  4. 

E.g.,  to  reduce  ^x^y/2a^b  to  an  equal  fraction  with  denom- 
inator "^a^bc, 

then  2a^bc  :  2a^b  =  aCs  and  the  fraction  sought  is  2acx^y /2a^bc. 

So,  to  reduce  2x^z/?>a^c  to  an  equal  fraction  with  numerator 
Qx^yz, 

then  QT^yz  :  2a;^2=3«/,  and  the  fraction  sought  is  Qa?yz/^a^cy, 

So,  entire  and  mixed  numbers  are  reduced  to  simple  fractions. 

E.g.,  x-^2a  =  {dx  +  2ad)/d,     x-1a-\-a^/d-(dx-'Xad^ie)/d, 

PrOB.  7.    To  REDUCE  TWO  OR   MORE    SIMPLE  FRACTIONS  TO 
EQUAL   FRACTIONS    HAVING   A    COMMON    DENOMINATOR. 

Over  the  continued  product  of  the  denominators  write  the 
product  of  each  numerator  into  all  the  denominators 
except  its  own,  or  [I  tli.  2  cr.  4. 

find  some  number  that  can  he  exactly  divided  hy  all  the  de- 
nominators; 
divide  this  number  by  the  denominators  in  turn  and  nudtiply 
each  numerator  by  the  quotient  got  by  using  its  denom- 
inator as  divisor. 

5^y      3^      Z(a-b) _  35rA/       A^ahc      Qaxja-b) 
^^•'    2a'      x'  7       ~   Uax'      Uax'         Uax     ' 

For  finding  the  lowest  common  denominator,  see  IV,  pr.  5o 


6,6.7,§5J  FRACTIONS.  65 

QUESTIONS. 

Reduce  tlie  fractions  below  to  lower  terms: 

a^-3x-^2  ^    a;^-2.y-15  aca^-h(ad-ic)x-^hd 

,    a^-y  ^         a^-V  ^        4x^-9 


a''±2abi-b^'         '  4:x'±l2x  +  9' 

'*  (2.T  +  3?/)2-1622'  •      {'dx'  +  x-lf-(x'-x-3y    • 

m'-?c'  p^-q^  7^-^  x'^-y^ 

Change  the  fractions  below  to  equal  fractions: 

13.  x/(x  —  3)     with  denominator    x^  —  5x  +  6. 

14.  (4rt-3)/(4a-4)    with  denominator     16a^-2Sa-\-12, 

15.  («-Z»)/(rt"-H«"-'Z>4-  •  •  •  +^'^-')  with  denominator «"-Z>^ 

16.  {x  —  b)/(r^  —  l)     with  denominator     l—x\ 

17.  {b-a)/(2x+3){3-2x)    with  denominator    4.^2-9. 

18.  a/{a-c)(b-c),     b/{a-c){c-b),     c/[d'-{a^-b)c  +  ah'\ 

with  denominator     {b  —  c){c  —  a), 

19.  Reduce  the  fractions  below  to  equal  fractions,  with  the 

common  numerator     a^  —  b^: 
(i'-W        a?  +  b^        a^  +  a^  +  ab'-^b^       a^-a^  +  ab^-b^ 
a^^W'      a^-b^'      a^-a^b^ab^-b^~'      a^  +  a'b  +  ab^  +  b^' 
Reduce  to  equal  fractions  having  common  denominators: 

'^^'  \-x^'     (1-^)"     (1  +  ^f     "  '  a^-^ax'    x^-ax'    ¥^^ 
„^       a  3a          2ax  2  3  22:-3 

^'^*    Z Zf      TTZj      72 — Z:z*  '^'^'    ZTf 


24. 
25. 

26. 


a-x'    a-hx'    a^-x?'  '  x'    2x-V    4a;'- 1* 

a^  —  bc  y^  —  ca  (?  —  ab 

~{a^^{a-cy     (b-c){b^'     (c-a)(c^)' 

2_ 5  3 

{x-l){x-2y     x^-5x-^6'     .^.2_4^r  +  3' 

r^-r-2      r^  +  r-1       ?-=^-l       r^  +  l      1-r      r  +  1 


66  THE   PRIMARY    OPERATIONS   OF   ALGEBRA.     [  II,  Pas. 

PrOB.  8.    To   ADD   FRACTIONS. 

Reduce  the  several  fractions  to  equal  fractions  having  a  com- 
mon  denominator;  [pr.  7. 

write  the  svm  of  the  7iew  numerators  over  the  common  de- 
nominator, 

U(?    ^{a-l)_  2\1&-V^ax{a-V) 
^^''     tax'^      7       ~  \^ax 

Subtraction  is  but  a  case  of  addition;  add  the  opposite  of 
the  suhlrahend. 

Uc^ _ 3{a-l)) _  2Uc^-(iax(a-b) 
^^''     2ax  7       ~  Uax 

PrOB.  9.    To   MULTIPLY    FRACTIONS. 

W?'ite  the  prodtict  of  the  numerators  over  the  product  of  the 
denominators,  cancelling  any  factor  that  is  common  to 
a  mwierator  and  a  denominator.  [I  th.  2  crs.  2,  4. 

^bf     ?>(a-b)  _  %c\a-b)^    Uf     Say  _  ig 
^^''     2ax^~     7       ~       Uax     '    2ax^  90^c^~  3bx 


Division  is  but  a  case  of  multiplication  ;  multiply  by  the 
recijirocal  of  the  divisor. 


E.g.. 


Uc^     3(a-b)_^bc'  7        _        7^6-^ 


2ax  '        7  2ax     'd(a  —  b)      2ax{a  —  by 


A  complex  fraction  is  an   indicated   division  wherein  the 
dividend,  the  divisor,  or  both,  are  simple  fi-actions. 

E.g., 


a;2_y2        ^2  +  _^2_       (.,;2_^/2).(.,.2_|_^/2) 

x'-y" 

x-\-y     x-y    ~    (x -\- yf - (x - y)'^ 

Axy 

x-y    x+y         {^-y)-{x+y) 

x?-f 

__  4icy     x^-y^  _     xy 
~  x^-y'^      ^xy  ~  x?-\-\f 

This  example  may  also  be  worked  by  multiplying  both 
terms  of  the  complex  fraction  by  x^  —  y*. 


B,  9,  §5] 


FRACTIONS. 


QUESTIONS. 
Add,  subtract,  multiply,  and  divide  as  shown  below: 


1.  1 


1 


:  1  +  .6-    1 


2.   1 


J^ 

1  +  x     1 


1 


„    l-x     1+x 


a  +  b     a  —  h     a-\-b     a—b 
a  +  x     a  —  x     a  —  x     a  +  x' 


1 


a-vb     a^-b''     a^  +  b' 


V. 


8. 


(a-b)(a-c)       {b-c){b-a)     {c-a)[c-b)' 
cr  b^  & 


{a-b)'{a-c)     (b-c)-{b-a)     {c-a)-{c-b)' 


^n 


1     a;"  +  l     r^"-l     x""  ^\ 


9.  i  +  _^  +  i^  +  ^ 
ij-Vz     y-z     z-y 


11. 
12. 
14. 

15. 
16. 

18. 


a  +  1 


+ 


^  +  1 


+ 


c  +  1 


a{n-b){a-c)     b{b-c){b-a)     c{c-a){c-b) 


M'4)<-a^(>4)- 


ij     a-b    x  +  y' 

^ ^  x^-2bx^  +  b~x^ 

^bx  +  b"'    x-b  '  x^^bF  '  "  x^-bx-^W  '' 
X?  a^  —  x^  a  —  x  a^  —  ax  +  x^   a^-\-2ax  +  x^ 


a^  +  x^  a^-\-:i^  a  +  x  a^-\-ax-\-x^   a^  —  2ax  +  x^' 


x  +  1  y^  +  2y  +  l   {x-l)y 
a 


b         c  abc 


IT. 


a  +  b    b^-a^  (l 


m         in^ 
a^-ab  +  b^ 


\x     a  —  b] 


c'-b' 


b  +  c  a  +  c  a^b  '  (a^  +  b^)(a^-{-c^) 

20.  Reduce  the  complex  fractions  below  to  simple  fractions: 
x  —  y  1)}^  +  w  n -\- n^ 

1  +  xy  Vi^  +  n^ 


if 


u\x-yy 
l  +  xy 


vr—  )r 


7ir  —  mn-{-7r 


p'  +  q' 
P'  +  'f. 

m  +  71     m  —  n 
m  —  n   '  m  +  n 

if-(r 

m  —  n     m  4-  7i 

p'-f 


m  + 11     m  — 11 


68  THE  PRIMARY  OPERATIONS  OF  ALGEBRA.  [U* 

§6.   QUESTIONS  FOR  REVIEW. 

Define  and  illustrate : 

1.  An   algebraic   expression;   a   binomial;   a   trinomial;   a 
quadrinomial;  a  polynomial. 

2.  Expressions  that  are  literal;  numerical;  entire;  fractional; 
symmetric;  homogeneous. 

3.  A  series;  a  finite  series;  an  infinite  series. 

4.  The  degree  of  a  term,  and  of  a  polynomial;  a  coefficient; 
like  terms;  unlike  terms. 

Give  the  general  rule,  with  reasons  and  illustrations,  for: 

5.  Adding  like  numbers;  unlike  numbers. 

6.  Subtracting  one  number  from  another. 

7.  Multiplying  a  monomial  by  a  monomial;  a  polynomial 
by  a  monomial;  a  polynomial  by  a  polynomial. 

8.  Dividing  a  monomial  by  a  monomial;  a  polynomial  by  a 
monomial;  a  polynomial  by  a  polynomial. 

9.  Reducing  a  simple  fraction  to  lower  terms;  to  an  equal 
fraction  having  a  given  numerator  or  denominator. 

10.  Reducing  two  or  more  simple  fractions  to  equal  frac- 
tions having  a  common  denominator. 

11.  Adding  and  subtracting  fractions. 

12.  Multiplying  and  dividing  fractions. 

State  the  principles  that  relate  to  the  form  of  a  product: 

13.  As  to  its  independence  of  the  values  of  the  letters. 

14.  As  to  its  symmetry. 

15.  As  to  the  sum  of  its  coefficients. 

16.  As  to  the  degree  of  its  highest  and  lowest  terms. 

17.  As  to  its  homogeneity. 

18.  As  to  the  number  of  terms.  . 


19.  Write  down  the  most  useful  type-forms. 

20.  State  what  arrangement  of  terms  is  best  in  multiplica- 
tion; in  division. 


$6]  QUESTIONS   FOR  REVIEW.  69 

21.  Explain  cross  multiplication;  and  show  how  it  is  used 
in  multiplying  numerals. 

22.  Explain  the  use  of  detached  coefficients,  in  multiplica- 
tion; in  division. 

23.  Show  how  the  symmetry  of  the  factors  helps  to  deter- 
mine the  product;  the  quotient. 

24.  Explain  the  methods  of  contraction  in  multiplication; 
in  division. 

25.  Explain  the  checks  used  in  multiplication;  in  division. 

26.  Explain  synthetic  division. 

27.  Draw  a  line  whose  length  equals  the  sum  of  two  given 
lines,  and  show  by  a  diagram  that  the  square  on  this  line  is 
made  up  of  a  square  on  each  of  the  two  given  lines  and  two 
rectangles  having  these  lines  as  sides:  hence  illustrate  the 
formula     {a  +  'by  =  (i'  +  2ah  +  b\ 

So,  the  formulae  {a  -  hf  =^  a"  -  2ah  +  h\  {a  +  b).(a-I))  =  a^ -  h\ 

28.  How  might  the  knowing  that  the  product  of  homoge- 
neous factors  is  homogeneous  help  to  find  errors  in  division  ? 
Add,  and  arrange  the  sum  to  falling  powers  of  x  and  the 

coefficients  to  falling  powers  of  y, 

29.  x*'  +  ^xy^ -  ^xz^  +  ix^y  -  4:xh  +  Qx^y"^  +  6.rV _  i2xyh 
+  Uxyz^  - 122:^^^  +  ?/*  -  4:yh  -  iy:^  +  z^-\-  6«/V  _,_  ^i^ 

Expand  and  add : 

30.  (a  +  b-\-cy+{a-\-b-cY+(a-b^cy-^{-a-\-b  +  cy, 

31.  {a  +  b-hcy  +  (a-^b-cy-\-(a-b-hcy^(-a  +  b  +  cy, 

32.  ia  +  b  +  cY-\-{a  +  b-cy-\-(a-b  +  cy  +  (-a-\-b-hcy, 
Given     a-{-/3—  —b/a,     aft  —  c/a\     then: 

33.  a^^^^{a^^y-'lap^bya^-1c/a^{b''-'lca)/a^, 

34.  a' ^ ^^^{a\ ^y -ZaP{a^ ^)^{-b^ ^Zabc)/w'. 

35.  a'^  +  «^/5*  ^-{W-  3«c)  •  bc^la\ 

36.  1/^  +  1//?= -V^.  37.   a/l3  +  p/a=[b''-2ca)/ca, 

38.  l/a^  +  \/fS^=  (b^  _  ^ca)/c\ 

39.  {a/p-^/ay  =  b\b^-iac)/ah\ 


70  SIMPLE  EQUATIONS. 

III.    SIMPLE  EQUATIONS. 


A  n  equation  is  a  statement  that  two  expressions  are  equal. 
These  two  expressions  are  the  members  of  the  equation. 

An  identity  is  an  equation  that  is  true  for  every  value  of 
the  letters  involved;  the  sign  of  identity  is  — . 
E.g.,  {x  +  a)  X  {x  —  a)=x^  —  a^    for  all  values  of  x  and  «; 
but  the  equation     5ic  +2  =  17     is  true  only  when  x  is  3, 
and  the  equation     x^  —  ^x^A.     is  true  only  when  a;  is  4  or  —1. 

The  letter  or  letters  whose  values  are  sought  are  the  un- 
knoiun  ele7nents;  the  other  elements  are  knoion  elements.  The 
unknown  elements  of  an  equation  are  usually  represented  by 
the  last  letters  of  the  alphabet;  and  in  literal  equations  the  first 
letters  then  stand  for  known  elements. 

The  solution  of  an  equation,  or  set  of  equations,  consists  in 
making  such  transformations  as  shall  result  in  giving  the 
values  of  the  unknown  elements.  The  values  so  found  are 
the  7'oots  of  the  equation  or  set  of  equations;  and  the  test  to 
be  applied  to  them  is  to  replace  the  unknown  elements  by 
these  values,  and  see  whether  they  make  the  equations  true. 
E.g.,  of  the  equation     2x  =  4:,     [x  unknown],  2  is  a  root, 

•.•2.2  =  4.  [df.  root. 

So,  of  the  equation  2:^  —  5.^  +  6  =  0,  [.t  unknown],  2,  3  are  roots, 
-.•22-5-2  +  6  =  0,     3^-5-3  +  6  =  0. 

The  degree  of  an  equation  is  that  of  its  highest  term. 

If  the  unknown  element  enter  an   equation   by  its  first 
power  only,  the  equation  is  a  simple  equation. 
E.g.,  2x  =  4:,  [x  unknown]  is  a  simple  equation. 

AXIOMS   OF    EQUALITY. 

1.  Numlers  equal  to  the  same  number  are  equal  to  each  other, 

2.  If  to  equal  numbers  equals  be  added,  the  sums  are  equal. 

3.  If  from  equal  numbers  equals  be  subtracted,  the  re 
MAINDEKS  are  cq2iaL 


AXIOMS  OF   EQUALITY.  71 

4.  If  equal  nuiiibers  le  vmUipUed  by  equals,  the  products 
are  equal. 

5.  If  equal  mimders  be  divided  by  equals,  the  quotiexts 
are  equal, 

6.  If  equal  numbers  be  raised  to  like  integer  powers,  the 
POWERS  are  equal. 

7.  If  of  equal  numbers  like  roots  be  taken,  the  roots  are 
equal. 

questions. 

1.  Show  that  7  is  not  a  root  of  the  equation  r?;  — 4  =  2. 
What  number  is  a  root  of  this  equation  ? 

2.  Write  an  equation  that  is  not  simple. 

3.  Show  that  ~3  is  a  root  of  the  equation  rc^  =  9. 

What  other  root  has  this  equation  ?    are  the  two  roots  equal? 

4.  What  is  the  difference  between  an  axiom  and  a  theorem? 

5.  If  from  each  member  of  the  equation     a:4-5  =  ll  — 2.r, 
6  be  subtracted,  what  term  in  the  first  member  is  cancelled  ? 

What  change  is  made  in  the  second  member? 

So,  if  2^  be  added,  what  change  is  made  in  each  member  ? 

6.  May  the  equation  x-\-a  —  b  —  1x  be  written  a;  +  2a:  =  ^  — «? 
How,  then,  may  a  term  be  transposed  from  one  member  of 

an  equation  to  the  other,  and  by  authority  of  what  axioms  ? 

7.  In  the  equation  \x  —  \x-\-\,  by  what  single  number  may 
the  two  fraction-tei-ms  be  multiplied  so  as  to  become  integers? 

What  other  term  must  then  be  multiplied  by  the  same 
number,  and  for  what  reason? 

8.  What  axiom  gives  authority  foi  changing  the  signs  of  all 
the  terms  of  an  equation? 

9.  AVhen  from  the  equation     5x  =  12,     we  get  the  value  2| 
for  X,  what  axiom  is  applied  ? 

So,  when  from  the  equation    sjx  —  '^,     x—^     is  found  ? 

10.  Write  an  equation  of  the  second  degree  with  x,  y  as  un« 
known  elements;  so,  of  the  third  degree. 


72  SIMPLE  EQUATIONS.  [in,  Pr. 

§1.  ONE  UNKNOWN  ELEMENT. 

PROB.    1.    To   SOLVE  A   SIMPLE   EQUATION",   OKE   UNKNOWN 
ELEMENT. 

Multiply  both  memlers  of  the  equation  hy  some  number  that 
contains  as  factors  all  of  the  deji07ni)iators,  if  any; 

[ax.  4. 
transpose  to  one  member  all  terins  that  involve  the  unknow7i 
element  and  to  the  other  member  all  other  terms; 

[axs.  3,  3. 
reduce  both  members  to  their  simplest  for^n; 
divide  both  members  by  the  coefficient  of  the  unlcnoiun  element, 

[ax.  5. 
Check.  In  the  original  equation,  replace  the  unh7iown  elemeiit 

by  the  result  so  found. 
E.g.,  if  ^{x  + 12)  =  1(6  +  32:)  -  \x\  [x  unkn. 

then  •••  7a;  f  84  =  36  +  ISx  -  1x,  [mnlt.  by  42. 

and       7a;  -  18a;  +  7a; = 36  -  84,  [trans.  84,  18a;,  -  7a;. 

.%  —4a;  =—48,     and    a; =12;  [div.  by  —4. 

and  vi(12  +  12)=|(64  36)-2,  [repl.  ii;  by  13, 

/.  12  is  the  root  sought. 

If  the  known  elements  be  wholly  or  in  part  literal  the  pro- 
cess is  essentially  the  same. 
E.g.,  if  ax  -  (Z>a;  + 1  )/x  -  a(x^  - 1 )/% 

then     ao!^—bx-l  —  ax^-a,  [mult,  by  .r. 

ax^ — ax^ ~bx=  —a  +  l,  [trans,  ax^,  —  1 

bx=a  —  l,  [cancel  ax^^  div.  by  "1. 

x={a-l)/b.  [div.  byZ> 

So,  if    (a-b)(x-c)-(b-c)(x~a)-(c~a){x~b)^0, 

then     ax  —  bx  —  ac  -\-  be  —  bx  -\-cx  -\-  ab  -  ac  —  ex  +  ax -\- be  —  ab  =  Oy 

2ax-2bx  =  2ac-2bc, 

x=c,  [div.  by  2(rt-Z>). 


1,§1]  ONE   UXKXOWX   ELEMENT.  73 

QUESTIOXS. 

Find  values  of  x  that  make  true  equations  of  the  statements: 
1.  l^-bx  =  V6-x,  2.  l-5x  =  Ta:  +  3.  3.  3x  +  6-2x  =  7x, 
4,  8  +  4^;  — 12:c  — 16.  5.  a  —  2x  —  x  —  b.  6.  7n  —  7ix=px-{-q. 
7.  22;  +  K4  +  ^)  =  3f.  8.  4:X-l  =  ix  +  2x-hi. 

9.  {x  +  ^)(x-2)  =  {x-d){x-d).     10.  {x  +  l){x-l)  =  x{x-2), 

Ih  (x-2){x-li)  +  {x  +  l){:c-d)-8x  =  2{x-8){x-7)-2, 

12    -^+-1-^_1-      13    ^^U     14    ^^-3_  6^  +  5 

•  a-  +  l^a;  +  2      a:  +  3'         '  x-2  3x  +  4:~  9^'-10* 

a;-a  _  3x-G  2x-5  _  6x  +  d  1_^     ^_  3 

'  2x-b~  6x-d'        '2^70;     2^'^7i~28' 

18.  i{7x  +  5)-i{o-x)^7i-ix-i{8-rx). 

19.  l(.,  +  lO)-|(3.r-4)+l(3.^'-2)(2.^:-3)  =  .^^-3-V 

/^' 1     .^ 2      i?;  —  3     X 4 

20.  ^^ — ^  — ' ^=  '-^ — j~- i.     [simp,  each  mem.  sejiarately. 

X       rj        X       O         X       'i       ^       0 

^.        1  1     _     1 ]_    ^.^   a:-4     a:-5__a;-7     a; -8 

'^  '  x  —  3     x  —  4i      x  —  b     x  —  Q''^'^'x  —  5     x  —  Q~x  —  8     x  —  d° 
23.  i(x-ia)+i{^-ia)+i(x-ia)  =  0. 


24.  24-2;-[7:r-{8a;-(9.r-3.?;-62-);]^0. 

25.  (.7:  +  5)2=(4-af +  2L^.      26.  Ct-?>0('^  +  ^0  =  ^(^-^). 

^^    :c  — 1     a;  — 2      2  ^^      aa;^  r/a:     ^ 

27. jr  =  r*  28.  T +  rt  +  — =  0. 

a  0        0  o  —  cx  c 

29.  «a:-J(:^-l)-c  =  0.  30.  («^  -  :r)(«H  »:)  =  «*  + 2 ^rr-a:^. 

31.  {x  +  iy^x[Q-{l-x)'\-2,  32.  C'i:-l)'-=Gr-2)(.7;  +  3). 

33.  (5a;  -6)/w-(a;-l)/?i  =  2^-2.     34.  px-qx={p  +  q)x- q\ 

35,  i(a;  +  4)-i(a;-4)=2  +  J^(3x-l). 

36.  i(3.^  +  2)-i(12-a;)  =  irr.   37,  {x  +  a){x-h)  =  (x-c){x^  d). 

38.  «a;  —  m  —  2 { ^ri;  —  ?j  —  3[ca;  — jt?  —  4:(dx  —  q)^\=  0. 

39.  x-[2x  +  {3x-^x)'\-bx-\Qx-{{7x  +  8x)-^x]\  =  -Z0, 

40.  i(a;-3)(2;  +  2)-i(a;-4)(2a;  +  l):z.7. 


74  SIMPLE  EQUATIONS.  [III.Pr, 

SPECIAL  PROBLEMS. 

Note  1.  If  the  statement  of  the  problem  be  in  words,  that 
statement  must  be  first  translated  into  algebraic  form. 
E.g.,  to  divide  $6341  among  A,  B,  C,  so  that  B  shall  have 

$420  more  than  A,  and  0  |560  more  than  B: 
put  X  for  A's  share,  ic  +  420  for  B's,  a; +  420 +  560  for  C's; 
then  •/ a;+^+420  +  ^+420T560  =  6341, 

.•.3a;=  6341 -420 -420 -560  =  4941,     and     x  =  lQ^l', 
/.  A  has  $1647,  B  has  $2067,  C  has  $2627. 

So,  to  divide  144  into  four  parts,  such  that  the  first  part  in- 
creased by  5,  the  second  decreased  by  5,  the  third  multi- 
plied by  5,  and  the  fourth  divided  by  5,  are  all  equal: 

put  X  for  the  number  to  which  the  several  results  are  equal; 

then  ','x  —  b+x-{-b-\-X'.b-{-x-b  =  144, 

.-.  5rz;  -  25  +  ^x  +  25  +  a:  +  25a;  =  720,  [mult,  by  5. 

i.e.,       36a;  =  720,     a;  =  20, 

and  the  parts  are  20-5,  20  +  5,  20  : 5,  20-5;  i.e.,  15,  25,  4, 100. 

So,  to  find  a  number  such  that  if  5,  15,  35  be  added  to  it,  in 
turn,  the  product  of  the  first  and  tliird  sums  shall  be 
10  more  than  the  square  of  the  second  : 
put  X  for  the  number,  a;  +  5,  a;  +  15,  a;  +  35  for  the  three  sums; 
then  V  (a; +  5)  X  (a;  +  35)  =  (a;  +  15f +  10, 
.-.  a;2  +  40a;  + 175  =  a;^  +  30a;  +  225  + 10, 
.*.  10a;=60,    a;  =  6,     and  the  numbers  are  11,  21,  41. 

So,  the  width  of  a  room  is  two  thirds  of  its  length;  if  the 
width  were  three  feet  more  and  the  length  three  feet 
less,  the  room  would  be  square;  find  its  dimensions; 
put  X  for  a  side  of  the  supposed  square; 
then  the  length  of  the  room  is  a; +  3  and  the  width  a;  — 3, 
and  •.•a;-3  =  |(a;  +  3), 

.*.  3a;  —  9  =  2a;  +  6,    a;=15,    and  the  room  is  12  by  18  feet. 
In  solving  problems,  it  is  not  sufficient  that  the  result  found 
shall  satisfy  the  equation:  it  must  also  satisfy  the  conditions 
of  the  problem  as  expressed  in  words. 


J 


i,§i]  ONE  UNKNOWN  ELEMENT  75 

QUESTIONS. 

1.  If  to  the  double  of  a  certain  number  14  be  added,  the 
gum  is  154:  what  is  the  number? 

2.  If  to  a  certain  number  46  be  added,  the  sum  is  three 
times  the  original  number:  find  the  number. 

3.  The  sum  of  two  numbers  is  20,  and  if  three  times  the 
smaller  number  be  added  to  five  times  the  larger,  the  sum  is 
84 :  what  are  the  numbers  ? 

4.  Divide  46  into  two  parts  such  that  if  one  part  be  divided 
by  7  and  the  other  by  3,  the  sum  of  the  quotients  shall  be  10. 

5.  In  a  company  of  266  men,  women,  and  children,  there 
are  four  times  as  many  men  and  twice  as  many  women  as 
children:  how  many  men  are  there?  how  many  women? 
and  how  many  children? 

6.  Thirty  yards  of  cloth  and  forty  yards  of  silk  together 
cost  866;  the  silk  is  worth  twice  as  much  per  yard  as  the 
cloth:  find  the  cost  per  yard  of  each  of  them. 

7.  My  purse  and  the  money  it  contains  are  together  worth 
$20,  and  the  purse  is  worth  a  seventh  part  of  the  money:  how 
much  money  does  the  purse  contain  ? 

8.  A  shepherd  being  asked  how  many  sheep  he  had  in  his 
flock,  said  "  if  I  had  as  many  more,  half  as  many  more,  and  7 
sheep  and  a  half,  I  should  then  have  500":  how  many  sheep 
had  he  ? 

9.  A  is  58  years  older  than  B,  and  A's  age  is  as  much  above 
60  as  B's  age  is  below  50 :  find  their  ages. 

10.  What  number  is  that  whose  double  being  added  to  24, 
the  sum  is  as  much  above  80  as  the  number  itself  is  below  100? 

11.  What  number  is  that  from  which  if  5  be  subtracted, 
two  thirds  of  the  remainder  is  40  ? 

12.  A  and  B  together  can  do  a  piece  of  work  in  8  days,  A 
and  C  in  9  days,  B  and  C  in  10  days:  in  how  many  days  can 
each  man  do  the  work  alone  ?  in  how  many  days  can  they  do 
it  all  working  together  ? 


76-  SIMPLE  EQUATIONS.  [in.PR. 

*  GENERAL  FORMS. 

Note  2.  Every  simple  equation  with  one  unknown  element 
may  be  reduced  to  the  form  ax-^h  —  a'x-^-'b',  whose  solution 
gives  x={b'  —  b)/{a  —  a'),  a  value  that  may  be  positive,  nega- 
tive, zero,  infinite,  or  indeterminate,  according  to  the  relations 
between  the  elements  a,  a',  h,  b']  and  there  are  three  cases: 
(a)  a^a*\  then  x  has  a  single  value,  positive,  negative,  or 

zero,  that  satisfies  the  equation. 
(J)  a  —  a'f  h^^V',  then  a;=  go,  wherein  oo,  read  infinity,  de- 
notes a  number  larger  than  can  be  named. 
This  result  may  be  interpreted  by  saying  that  if  a  and  a',  or 
either  of  them,  take  gradually  changing  values,  and  if  a  be 
not  equal  to  a'  but  approach  nearer  and  nearer  to  a'',  then  x 
grows  larger  and  larger  without  bounds. 
E.g.,  if  A,  A'  travel  along  the  same  road  in  the  same  direction 
at  a,  a'  miles  an  hour,  and  if  A  be  now  h  miles  and 
A',  h'  miles  from  the  same  starting  point;  then  the 
quotient  {b'  —  h)/{a  —  a')  is  the  time  that  will  elapse 
before  they  come  together. 
If  the  hourly  gain,  a  — a',  be  small,  that  time  is  long; 
if  there  be  no  gaiil,  i.e.,  if  a  —  a\  they  will  never  be  together, 
and  there  is  no  finite  value  of  x  that  satisfies  the  equa- 
tion, 
(c)  a  =  a',  2' =  5';  then    a;  =  0/0,    and  the  equation  is  satisfied 
by  any  number  whatever. 
These  cases  may  be  further  illustrated  by  this  question: 
Two  men  A,  A'  have  b,  b'  dollars  and  save  a,  a'  dollars  a 
year:  in  how  many  years  will  thoyhave  the  same  assets?    The 
interpretation  of  the  principles  in  terms  of  the  problem  is  this: 
If  h*>b,  the  time  sought  is  in  the  future  if  aya'j  but  in  the 

past  if  a<a\ 
and  if  b* <b,  these  results  are  reversed; 
if  h^b'y  a'  =  a,  the  answer  is  never'; 

if  b=-h\  the  present  is  the  time  sought  if  a^a',  but  if  a=a', 
the  two  men  have  always  the  same  assets. 


1,§1)  ONE   UNKNOWN   ELEMENT.  T7 

QUESTIONS. 

1.  If  tlie  equation  x  —  20=—3x  be  represented  by  the 
general  formula  ax-{-b  =  a^x  +  b',  for  what  number  does 
each  of  the  letters  a,  b,  a',  b' ,  stand  ? 

2.  In  the  fraction  {b'  —  b)/{a  —  a')  what  is  the  sign  of  the 
denominator  if  «>«'?  \i  a<a''^ 

So,  if  b'>b,  what  is  the  sign  of  the  numerator?  if  b' <bt 

3.  What  is  known  about  the  value  of  ^  if  ^'>Z'  and  «>«'  F 
So,  if  V>b  and  a<a'  ?  if  V <b  and  a>a'  ?  if  ¥ ^b't 

4.  What  is  the  value  of  a  fraction  whose  numerator  is  zero  ? 
Show  that  this  value  multiplied  by  the  denominator  gives 

the  numerator,  and  that  no  other  value  will  give  it. 

5.  Reduce  the  fractions  6/3,  6/.  3,  6/.  03,  6/.003  •  •  •  to 
whole  numbers :  what  change  is  going  on  in  the  series  of  de- 
nominators and  what  in  the  quotients?  if  the  denominator  be 
very  small,  what  is  the  quotient?  if  the  denominator  be  0  ? 

6.  How  is  an  example  in  division  proved? 

Prove  that:  0/0  =  2;  0/0  =  10;  0/0  =  5000;  0/0=  -  .12. 
What  is  the  value  of  {b'  —  b)/(a  —  a')  when  b'  —  b,  a=^a'  ? 
What  is  meant  by  an  indeterminate  expression  ? 

7.  In  the  case  of  the  two  travelers  A,  A';  if  a  —  a\  b  —  b', 
are  they  now  together  ?  how  long  have  they  been  together  ? 
how  long  will  they  remain  together  ? 

8.  If  b'>b,  a<a\  is  the  time  of  meeting  past  or  future? 
if^'>^,     a>a'?    if  b'<b,     a<a'?    if    b'<b,     a>a'? 

9.  What  is  the  meaning  of  the  problem  if  b,  b'  be  of 
opposite  signs  ?  both  negative  ?  if  a,  a'  be  of  opposite  signs  ^ 
both  negative  ? 

10.  A  gives  a  house  worth  b  dollars  and  land  worth  a  dollars 
an  acre  in  exchange  for  B's  house  worth  b'  dollars  and  as 
many  acres  of  land  worth  a'  dollars  an  acre:  how  large  is 
each  estate  ? 

Discuss  the  problem  for  the  different  relations  between 
«,  «',  b,  b'  considered  before:  which  of  the  results  interpieted 
in  the  last  question  on  page  7G  has  no  meaning  here  ? 


78  SIMPLE   EQUATIONS.  LU1,Pr. 

§2.   TWO   UNKNOWN    ELEMENTS. 
Equations  that  iuvolve  the  same   unknown  elements,  and 
are  satisfied  by  the  same  values  of  them,  are  simultaneous 
equations;  and  those  values  are  simultaneous  values. 
E.g.,  if  the  equations  2:e  +  5i/  =  19,  6a;  — 3z/  =  3,  \x,  y  unknown] 
be  simultaneous,  2,  '6  are  a  pair  of  roots, 
•.•2.2  +  5.3  =  19,     6.2-3.3  =  3. 
Elimination  is  that  process  by  which  an  unknown  element 
is  removed  from  a  pair  of  equations. 

PrOB.  2.    To    ELIMINATE   AN    UNKNOWN    ELEMENT   FROM   A 
PAIR   OF    SIMPLE    EQUATIONS. 

BY   ADDITION   AND   SUBTRACTION. 

Find  some  number^  as  small  as  may  be,  that  exactly  contains 

both  the  coefficients  of  the  element  to  be  eliminated; 
divide  this  number,  in  turn,  by  these  coefficients,  cmd  multiply 
the  two  equations  through  by  the  quotients;  [ax.  4. 

subtract  one  equation  from  the  oilier,  member  from  member. 
E.g.,  to  eliminate  x  from  the  pair  of  equations 

6.T  +  7^  =  85,      2:r  +  3?/  =  33: 
then  •.•  6  contains  6  once  and  2  three  times, 

.•.  Qx^ly  =  85,     62:  +  9z/  =  99,  [mult,  by  1,  3. 

.•.2?/ =  14.  [subtract. 

BY  COMPARISON. 

Solve  both  equations  for  the  element  to  be  eliminated;     [pr.  1. 
put  the  two  values  thus  found  equal  to  each  other.  [ax.  1. 

E.g.,  to  eliminate  x  from  the  same  pair  of  equations: 
then  2;  =  ^(85-7^)  =  K33-3?/),  [sol.  both  eq.  for  x. 

BY  SUBSTITUTION. 

Solve  either  equation  for  the  element  to  be  eliminated;     [pr.  1. 
in  the  other  equation,  replace  this  element  by  the  value  so  found. 
E.g.,  to  eliminate  x  from  the  same  pair  of  equations: 
then  •.•  rc  =  ^(33  -  3</),  [sol.  2d  eq.  for  x. 

.•.  99  -  9 «/  +  7^/  =  85.  [repl.  x  iu  1st  eq. 


2,  §3]  TWO   UNKNOWN    ELEMENTS.  79 

QUESTIONS. 

1.  Define  elimination;  what  is  the  derivation  of  the  word  ? 

2.  In  the  equation  2:6* +  5//=:  19  replace  x  by  —  |,  y  by  4: 
is  the  equation  true  for  these  values  ? 

Is  the  equation     6a:  — 3^  =  3     true  for  the  same  values? 
So,  in  the  second  equation  replace  x  by  3.  y  by  5:  do  these 
values  satisfy  the  first  equation? 

3.  Assume  any  value  at  random  for  y  in  2.t  +  5^  =  19:  can 
a  satisfactory  value  be  found  for  x  in  that  equation  ?  in 
6a:  — 3^  =  3?  with  the  same  value  of  y,  in  both  equations  at 
the  same  time  ?  what  is  needful  to  a  correct  solution? 

4.  What  two  axioms  are  applied  in  elimination  by  addition 
and  subtraction  ? 

5.  Multiply  the  equation  6.i-  +  7//  =  85  by  3,  2a: +  3//=!  33 
by  T;  then,  subtracting,  what  letter  is  eliminated  ? 

6.  In  eliminating  x  by  comparison,  how  is  it  known  that  the 
two  expressions  for  x,  found  from  the  separate  equations,  are 
equal  ? 

7.  To  get  definite  values  for  two  unknown  elements,  how 
many  independent  equations  must  be  used  ? 

8.  By  addition  and  subtraction,  eliminate  x  from  the  pair  of 
simultaneous  equations     5a;  +  6^  =  29,     3a'  +  2^  =  11. 

9.  So,  from     2.c  +  5^  =  23,     7a:  +  2y  =  34. 

10.  Eliminate  ?/ from     8a'  +  132/  =  79,     7a:-f2//  =  41. 

11.  By  comparison,  eliminate  x  from  the  pair  of  simul- 
taneous equations     4a:  — 3^=— 10,     7.1^  +  8?/ =  62. 

12.  So,  from     Ja:  +  4?/  =  18,     5a:-3?/  =  17. 

13.  Eliminate  y  from  2a:  +  4«/  =  20,     7a;  +  3j/  =  37. 

14.  So,  from  4/a:  +  7/i/  =  lf,  3/a:  +  5/?/  =  l|,  using  1/a:, 
1/y  as  the  two  unknown  elements. 

15.  By  substitution  eliminate  x  from  the  pair  of  simul- 
taneous equations     3a:  — 2^  =  1,    5a:  +  3?/  =  3:f. 

16.  So,  from     6a:  +  9^  =  15,     8a:-15^  =  ll. 

17.  Eliminate  ?/ from    \x-i^y=-^,     3,r  +  4?/  =  43. 


80  SIMPLE  EQUATIONS.  [III.Prs. 

THE  solutio:n"  of  simultaneous  simple  equations. 

Peob.  3.  To  solve  a  pair  of  simple  equations,  if  one 
has  two  unknown  elements  and  the  other  but  one. 

Solve  the  equation  that  has  hid  one  unhnoivn  element;   [pr.  1. 
rei)lace  this  element  hy  its  value  in  the  other  equation,  and 

solve  for  the  other  unknoivn  element,  [th.  4  cr.  2. 

E.g.,  to  find  X,  y  from  the  pair  of  simultaneous  equations 

6a;  +  ??/  =  85,     4?;  =  24: 
thena;  =  6,     36  +  7?/=:85,     7?/  =  49,     y  =  7. 
Prob.  4.  To  solve  a  pair  of  simple  equations,  if  both 

HAVE   THE   same   TWO    UNKNOWN   ELEMENTS. 

Combine  tJie  two  equations  so  as  to  eliminate  one  uiiknown 
element,  thus  forming  an  equation  involving  the  other; 
solve  this  equation  for  its  unhnoimi  element; 
rejolace  this  element  by  its  value  in  either  of  the  given  equations; 
solve  the  equation  so  found  for  the  other  unhnoivn  element. 
Check.  Repla-ce  tlie  tivo  unhnoivn  elements  by  their  values  in 
that  one  of  the  original  equations  which  ivas  not  used 
in  finding  the  value  of  the  second  element. 
E.g.,  to  find  X,  y  from  the  pair  of  simultaneous  equations 

6.T  +  7?/  =  85,     2x  +  3^  =  33: 
then  •.•  1(85  -  6a;)  =  ^(33  - 2.^'),  [elim.  y, 

.-.  255-183;  =  231-14:r,  [mult,  by  21. 

.-.  -4x=-24:,     x~G; 
.'.30 +  7?/ =  85,     y  =  7. 

DEPENDENT   EQUATIONS. 

Note  1.  If  one  of  the  two  equations  may  be  derived  from 

the  other,  there  is  no  single  solution,  but  an  infinite  number 

of  solutions.     The  equation  is  then  indeterminate. 

E.g.,  the  equations     2a:  +  3?/  =  13,     6.c  +  9?/  =  39     are  but  one 

equation  in  two  forms,  and  any  value  may  be  given  to 

either  of  the  unknown  elements,  and  the  corresponding 

value  of  the  other  computed. 


3,4,  §2]  TWO   UNKNOWN   ELEMENTS.  81 

QUESTIONS. 

Solve  the  pairs  of  equations  below,  and  check  the  work: 

1.  bx-?>y  =  VD,    2y  =  \^.  2.  3(a;  +  2^)  =  30,    |a:=3. 

3.  82'  +  3?/  =  14,    5j/  =  10.  4.  3a:-8^  =  7,    ^x  =  b, 

5.  2x-4-y  =  x-l,     -3^= -9. 

6.  2i  +  y-ix  =  iy,     fr  =  3i. 

7.  x-\-y  =  9,    x-y  =  l.  8.  15x-\-2y  =  17,    9x-iy  =  5. 
9.  5.r  +  3^?/  =  8,    7.T-3^  =  4.       10.  3x-\-y=:ie,    3y  +  x  =  8, 

11.  3y  =  DX,    l(jy  =  27x~l.         12.  8x  =  Dy,    Ux^8y  +  1. 
IS.  x  =  ^y,    x-^y  =  ^.  14.  lla;-3^  =  0,    x-y=-ie. 

15.  2x  +  y  =  0,    iy-Sx  =  8.         16.  x-y  =  ^,    .T  +  l  =  f. 


17.  '-t 
X      y 

1 
-6' 

7      3  _5 
X     y  ~  Q' 

'''l^r'*^ 

5  7  29 
x'^y-  12* 

19.  '-^ 
X     y 

"12' 

2      3           1 

a:      ?/  ~     12* 

oo    ^-^--1 
'•^•^     y-     6' 

3      1  _  1 

a;     3/  ~  30' 

21.  210.^ +  42j/  + 93  =  0,     22:c  +  14z/  +  7  =  0. 

22.  f2/-|a;  +  24  =  0,     f2/  +  ir  +  ll  =  0. 

g:  +  y  _        ^-3.?/     5?/-a;  _1 
'^*'  x-2y~'         6     "^     9      ~2' 
25.  V^(80  +  3.T)=:18i-i(4:^:  +  3?/-8), 
10?/  +  ^(6:c-35)  =  55  +  10.T. 

12^  +  7^_        18.y-7a;_  _^  ,  _1__9^    ^".V-i 

"''•    5.r-3^--'^     Qx  +  IQ-"^'  '^^'a  +  b^a-b~'^^'     4.ab-^' 

28.  l-^±^^=—         !l£.Z^^:::,3. 

x  —  yx  —  y*         23 

14  11 

29.  .^7^— r^+^T-^.-x  =  5, 


2(.r  +  l)  '  3(fy  +  l)       '      .T  +  1     3(^  +  l)~ 
30.  Write  down  any  simple  equation  at  will,  and  then  make 
dependent  equations  from  it  by  different  processes. 


82  SIMPLE  EQUATIONS.  f  m,  Pr 

SPECIAL   PROBLEMS. 

I^OTE  2.  In  solving  special  problems  it  may  be  convenient 

to  express  different  unknown  elements  by  different  symbols. 

E.g.,  a  vintner  at  one  time  sells  30  dozen  of  port  wine  and  30 
dozen  of  sherry,  and  for  tlie  whole  receives  $600;  and 
at  another  time  he  sells  30  dozen  of  port  and  25  dozen 
of  sherry,  at  the  same  price  as  before,  and  for  the  whole 
receives  $700 :  what  are  the  prices  ? 

pnt  X  for  the  price  of  a  dozen  of  port,  and  y  for  that  of  a 
dozen  of  sherry; 

then  •.•  Wx  +  30?/  =  $600,     30a;  +  25j/  =  $700, 
/.a;  =  $15,     ?/  =  $10. 

So,  if  a  certain  rectangular  bowling-green  were  5  yards  longer 
and  4  yards  broader,  it  would  contain  113  yards  more; 
but  if  4  yards  longer  and  5  yards  broader,  it  would 
contain  116  yards  more:  what  are  its  length  and  breadth  ? 

put  X,  y  for  the  length  and  breadth; 

then  '.•  {x  +  b)'{y  +  ^)  =  xy  +  ll'iy     (x  +  ^)'{y-^b)  =  xy  +  llQ, 
.-.  a;  =  12  yds.,     ?/  =  9yds. 

So,  if  the  number  of  men  engaged  upon  a  certain  piece  of 
work  be  made  5  greater,  the  work  can  be  done  in  4 
days;  if  5  less,  in  12  days:  how  many  men  are  at  the 
work,  and  in  how  many  days  can  they  do  it  ? 

put  X  for  the  number  of  men  and  y  for  the  number  of  days; 

then  one  man  could  do  the  work  in  xy  days, 

and   '.'  4L{x-\-b)  =  xy,     12{x  —  b)  =  xy, 

.\4{x  +  5)  =  l2(x-5),     iC^lO,    y  =  6. 

So,  a  certain  two-figure  number  is  G  greater  than  6  times  the 
sum  of  its  digits,  and  reversing  the  order  of  the  digits 
makes  the  number  less  by  3  times  its  first  figure;  find 
the  number: 

put  X  for  the  tens*  figure  and  y  for  the  units'  figure; 

then  •••  10x  +  y  =  G{x-\-y)  +  G,     '[0y  +  x=\0x-\-y-'3r, 
,\x=:d,     y  =  G,     and  the  number  is  96. 


4,  §2]  TWO   UNKNOWN   ELEMENTS.  83 

QUESTIONS. 

1.  Find  two  numbers  such  that  their  sum  is  27,  and  that, 
if  four  times  the  first  be  added  to  tliree  times  the  otlier,  tlio 
sum  is  93. 

2.  Find  two  numbers  such  that  twice  the  first  and  three 
times  the  second  together  make  189  and,  if  double  the  second 
be  taken  from  five  times  the  first,  7  remains. 

3.  A  flagstaff  is  sunk  in  the  ground  one-sixth  part  of  its 
height,  the  flag  occupies  6  feet,  and  the  rest  of  the  staff  is 
three-quarters  of  its  whole  length:  what  is  the  length? 

4.  The  diameter  of  a  five-franc  piece  is  37  millimeters  and 
that  of  a  two-franc  piece  27  millimeters;  thirty  pieces  laid  in 
contact  in  a  straight  line  measure  one  meter:  how  many  of 
each  kind  are  there  ? 

5.  A  certain  number  consisting  of  two  figures  is  equal  to 
four  times  the  sum  of  its  digits,  and  if  18  be  added  to  it  the 
order  of  the  digits  is  reversed:  what  is  the  number  ? 

6.  If  the  tail  of  a  fish  weigh  9  lbs.,  his  head  as  much  as  his 
tail  and  half  his  body,  and  his  body  as  much  as  his  head  and 
tail,  Avhat  is  the  weight  of  the  whole  fish  ? 

7.  There  are  two  pipes  one  of  which  will  fill  a  cistern  in  an 
hour  and  a  half,  the  other  in  two  hours  and  a  quarter:  in 
what  time  Avill  both  fill  it? 

8.  Divide  90  into  four  parts  such  that  if  the  first  be  in- 
creased by  2,  the  second  diminished  by  2,  the  third  multiplied 
by  2,  and  the  fourth  divided  by  2,  the  sum,  difference,  prod- 
uct, and  quotient  so  found  shall  all  be  equal. 

9.  A  and  B  engage  in  2)lay;  in  the  first  game  A  wins  as 
much  as  he  had  and  ^4:  more  and  finds  he  has  twice  as  much 
as  B;  in  the  second  game  B  wins  half  as  much  as  he  had  at 
first  and  $1  more,  when  he  has  three  times  as  much  as  A: 
what  sum  had  each  at  first  ? 

10.  What  fraction  is  that  which  becomes  equal  to  |  when 
its  numerator  is  increased  by  G,  and  equal  to  ^  when  its  de- 
nominator is  diminished  by  2? 


84  SIMPLE  EQUATIONS.  [Ih',Pr. 

DISCUSSION   OF   A   PROBLEM. 

Note  3.  To  discuss  a  problem  whose  answer  is  numerical 
is  to  try  whether  all  the  conditions  of  the  problem  are  satisfied 
by  all  or  any  of  the  numbers  that  are  found  to  satisfy  the 
equations  into  which  the  problem  was  translated ;  and,  if  not, 
to  observe  what  other  conditions  the  unknown  elements  must 
satisfy  besides  those  taken  account  of  in  putting  the  problem 
into  equation.  •  ; . 

To  discuss  a  problem  whose  answer  is  literal  is  to  observe 
between  what  limiting  numerical  values  of  the  known  ele- 
ments the  problem  is  possible;  whether  any  singularities  or 
remarkable  circumstances  occur  within  these  limits;  and  what 
changes  in  the  statement  of  the  problem  would  make  it  possi- 
ble for  the  other  values  of  the  known  elements. 
E.g.,  in  a  certain  two-digit  number  the  first  digit  is  half  the 
other,  and  if  27  be  added  to  the  number,  the  order  of 
the  digits  is  reversed:  what  is  the  number? 
put  X  for  first  digit,    y  for  second  digit; 
then-.-2:r=?/,     \i)x  +  y-{-21  =  lOy  +  x, 

.-.a;  =  3,     ?/  =  6,     the  number  is  36;     and  36  +  27  =  03. 

Were  this  the  statement:  in  a  certain  two  digit  number,  the 
first  digit  is  half  the  other,  and  if  24  be  added  to  the  number, 
the  order  of  the  digits  is  reversed; 
K\\Qia.\'2x  =  y,     10:c  +  ^  +  24  =  10?/  +  a:, 

.•.  ic=2|,     «/  =  5J,     and  the  number  is  impossible. 

The  statement  of  the  problem  puts  a  limitation  upon  x,  y 
not  expressed  by  the  equation:  they  must  be  integers. 

And  were  this  the  statement:  in  a  certain  two-digit  number 
the  first  digit  is  half  the  other,  and  if  a  be  added  to  the  num- 
ber, the  order  of  the  digits  is  reversed; 
then     ^x  —  y,     10rc  +  ?/  +  rt  =  10?/-4-a:,     a;  — «/9,     y  —  2a/^. 

Here  the  special  condition  is  imposed  that  a  shall  be  a  mul- 
tiple of  9  not  greater  than  36  nor  less  than  —36; 
i.e.,  a  is   36,  27,  18,     0,   0,     -9,  -18,  "27,  "36, 

and  the  number  is   48,  36,  24,  12,   0,     12,     24,  "36,  "48. 


4,§2i  TWO   UNKNOWN   ELEMENTS.  85 

QUESTIONS. 

1.  In  a  company  of  a  persons  each  man  gave  m  dollars  to 
the  poor,  each  woman  n  dollars;  the  whole  sum  was  ka 
dollars:  how  many  men  were  there  ?  how  many  women  ? 

Show  that,  if  m>ny  then  7n>k>n;  and  that  the  example 
is  possible  only  when  {m  —  'k)a,  (k  —  n)a  are  multiples  of  m  —  ?i. 

2.  A  is  a  years  old  and  B  b  years:  when  will  A  be  twice  as 
old  as  B  ?  What  relation  between  a  and  b  puts  the  date  sought 
in  the  future  ?  what,  in  the  present?  what,  in  the  past  ? 

3.  A  laborer  receives  a  dollars  a  day  when  he  works,  and 
forfeits  b  dollars  a  day  when  idle;  at  the  end  of  vi  days  he 
receives  Jc  dollars:  how  many  days  does  he  work,  and  how 
many  is  he  idle  ? 

What  relation  exists  between  a,  b,  k,  m,  if  his  forfeits  just 
cancel  his  earnings?  if  his  forfeits  exceed  his  earnings  ?  Give 
numerical  illustrations. 

4.  A  father  is  now  a  times  as  old  as  his  son ;  k  years  hence 
he  will  be  b  times  as  old:  what  are  their  ages  now  ? 

Give  numerical  values  to  a,  b,  k,  and  interpret  the  results. 
Show  that:    k>Oy  it  a>b;    ^'-0,  ita  =  b;    k<0,  \i  a<b. 

5.  The  sum  of  two  numbers  is  a,  and  the  difference  of  their 
squares  is  P;  what  are  the  numbers  ? 

Interpret  the  results:     ilk^>ci^;     it  k^  =  a^;     \tk^<a^. 

6.  The  difference  of  two  numbers  is  a,  and  the  difference 
of  their  squares  is  k~:  what  are  the  numbers? 

Interpret  the  results:     if  ^•^>«^;     itk'^  =  a'^]     if  ^'^<rtl 

7.  If  to  the  numerator  of  a  certain  simple  fraction  a  be 
added,  the  result  is  c/d,  and  if  to  the  denominator  a'  be 
added,  the  result  is  c' /d' :  what  is  the  original  fraction  ? 

8.  In  a  certain  two-digit  number  the  second  digit  is  a  times 
the  first,  and  if  b  be  added  to  the  number,  the  digits  are  re- 
versed:  show  that  a  may  not  exceed  9,  be  less  than  1,  or  be 
negative;  and  show  when  a  may  be  fractional. 

Show  that  b  is  a  multiple  of  9  or  of  a  — I',  and  show  what 
bounds  b  lies  between  for  different  values  of  a. 


86  SIMPLE  EQUATIONS.  [III.Pr. 

MORE    CONDITIONS  THAN   UNKNOWN   ELEMENTS. 

Note  4.  It  may  happen  that  the  problem  gives  more  condi- 
tions, and  so  more  equations,  than  unknown  elements;  such 
problems  can  be  solved  if  the  conditions  be  not  inconsistent. 
E.g.,  to  find  Xy  y  from  the  set  of  three  equations 

3a;  +  7?/  =  17,     bx-2y  =  l,     ac  +  ?/  =  10: 
take  the  first  two  equations  and  solve; 
then  '.' x  =  l,     j?/  =  2,     in  these  two  equations, 
and   •.*  these  roots  satisfy  the  third  equation, 

.*.  this  set  of  equations  is  possible,  and  the  roots  are  1,  2. 
But  not  possible  is  the  set  of  equations 

3^  +  7^  =  17,     bx-2y  =  l,     Sx  +  y^l'Z. 
fewer  conditions  than  unknown  elements. 
Note  5.  It  may  happen  that  the  problem  gives  fewer  con- 
ditions, and  so  fewer  equations,  than  unknown  elements;  such 
problems  are  indeterminate ,  and  the  set  of  equations  may 
have  many  sets  of  roots. 

E.g.,  to  find  x,  y  from  the  single  equation  2.c  +  3?/  =  12: 
put       y=.-.-4,     -3,     -2,-1,       0,  n,     +2,     +3,  +4,..., 
then     a;=..-+12,  +10i,  +  9,  +7i,  +6,  -^41,  +3,    +11,    0,    •  •  •, 
i.e.,  if  to  y  be  given  a  series  of  values  increasing  by  1,  there 

results  a  series  of  values  for  x  decreasing  by  IJ; 
orput^=...-4,     -3,    -2,     -1,       0,  n,     +2,     +3,  +4,    •  • -, 
then     2/=---"6|,  +6,    +5.^,  +4|,  +4,  +3^,  +2|,  +2,  n^,---, 
i.e.,  if  to  X  be  given  a  series  of  values  increasing  by  1,  there 
results  a  series  of  values  for  y  decreasing  by  f . 
If  either  of  the  unknown  elements  take  any  value  whatever, 
the  corresponding  value  of  the  others  may  be  found. 
E.g.,  if  x  =  4:^,     then  y  =  |;     or  if  //  =  4J,     x=  —  f. 

If  the  condition  be  imposed  that  the  roots  shall  all  be  in- 
tegers, or  all  positive  integers,  it  may  happen  that  tlie  equa- 
tions have  very  few  such  roots,  or  even  none  at  all. 
E.g.,  3,  2,  is  the  only  pair  of  positive  integer  roots  in  the  ex- 
ample above. 


4,  §2]  TWO   UNKNOWN   ELEMENTS.  87 

QUESTIONS. 

1.  Given  the  three  simple  equations 

2.^  +  3^  =  18,  3:c  — 2_y  =  l,  7:c  — 4^  =  5:  is  it  certain, 
before  solving,  that  values  of  x  and  y  can  be  found  that  will 
stitisfy  all  the  given  equations  ? 

Solve  the  second  and  third  equations  and  see  whether  the 
values  so  found  satisfy  the  first. 

2.  So,  for  the  three  equations 

3.  Find  two  numbers  whose  sum  is  60,  whose  difference  is 
24,  and  one  of  which  is  3  times  the  other. 

If  the  results  obtained  do  not  satisfy  all  three  conditions, 
show  what  change  in  each  condition  will  make  it  consistent 
with  the  other  two. 

4.  Solve  the  equations  6a;  — 8//  =  3,  15a;=7J  +  20?/,  or  ex- 
plain what  is  the  difficulty  with  them. 

5.  In  the  example  of  note  5,  why  does  x  decrease  when  y 
increases  and  increase  when  y  decreases  ? 

6.  In  the  equation  4x  — 5^  =  1,  if  increasing  values  be  given 
to  X,  will  the  corresponding  values  of  y  increase  or  decrease  ? 

If  the  values  of  x  increase  by  1,  how  do  the  values  of  y 
change?  if  the  values  of  y  decrease  by  1,  how  do  the  values  of 
X  change  ? 

By  what  integers  may  x  increase  or  decrease  so  that  y  also 
shall  change  by  integers  only? 

What  are  all  the  pairs  of  integer  roots  smaller  than  20? 

7.  Show  that  the  equation  xy  =  24:  may  have  an  infinite 
number  of  pairs  of  roots,  and  that  the  value  of  one  root 
grows  smaller  as  the  other  grows  larger. 

Show  how  the  relations  of  x,  y  differ  in  this  example  from 
their  relations  in  ex.  6. 

Discuss  the  equation  xy=:0. 

8.  How  many  pairs  of  roots  has  the  equation     x  =  ay? 
What  relation  have  the  values  of  x  and  of  y  when  a  is  posi- 
tive ?  when  a  is  negative  ? 


88  SIMPLE  EQUATIONS.  [III,Pr. 


§3.    THREE   OR  MORE  UNKNOWN  ELEMENTS. 
Prob.  5.   To   SOLVE  A  SET   OF   n   independe:n^t  simple 

EQUATIONS  THAT  INVOLVE  THE   SAME  11  UNKNOWN  ELEMENTS. 

Combine  the  n  equations,  two  and  two,  in  n  —  \  ways,  so  that 

each  equation  is  used  at  least  once,  arid  so  as  to  elinii- 

nate  the  same  unlcnoion  element  at  each  operation; 
therely  form  n  —  1  equations  iiivolving  the  same  n  —  1  un- 

knotcn  elements; 
so,  comhine  these   n  —  1  equations,   and  therely  form   n  —  2 

equations  involving  the  same  n  —  2  unknotvn  elements; 
and  so  on  till  there  results  one  equation,  involving  hut  one  un- 

knoiun  element; 
solve  this  equation,  and  replace  the  imhnoivn  element  ly  its 

value  in  one  of  the  tivo  equations  involving  ttvo  un- 

Jcnown  elements; 

solve  this  equation  for  the  second  urihnoivn  element,  and  re- 
place these  two  ele^nents  by  their  values  in  one  of  the 
three  equations  involving  three  unknown  elements; 

and  so  07i  till  all  the  roots  are  found. 

E.g.,  to  find  X,  y,  z  from  the  set  of  equations 

x  +  2y  +  ^z  =  14:,      3a;  +  2?/  +  2;=10,      6.r  + 9?/  + 132  =  03: 

then     6a: +  12?/ +  182  =  84         [4]  [mult,  first  cq.  by  6. 

6:c  +  4?/  +  22  =  20  [5]  [mult.  sec.  eq.  by  2. 

6a;  +  9i/  +  132  =  63  [6] 

By  +  162  =  64  [7]  [sub.  eq.  5  from  eq.  4. 

5y  + 112  =  43  [8]  [sub.  eq.  5  from  eq.  6. 

40?/ +  802  =  320  [9]  [mult.  eq.  7  by  5. 

40?/  +  882  =  344  [10]  [mulfc.  eq.  8  by  8. 

82  =  24     and     2=3  [sub.  eq.  9  from  eq.  10. 

8?/  =  G4-162  =  10     and     ?/  =  2. 
a;=14-2?/-32=l. 


5,  S3j  THREE   OR  MORE   UNKNOWN   ELEMENTS.  89 

QUESTIONS. 

1.  How  many  independent  equations  make  it  possible  to  find 
the  value  of  four  unknown  elements  ?  of  five  ?  of  ten  ?  of  n  ? 

Solve  the  sets  of  equations: 

2.  Zx-^  +  bz^^,     Sx-y-z  =  &,     '^x-by -'dz=  -I. 

3.  x-2y-bz  =  20,    3x-Dy-3z  =  22,     -8^  +  ll?/  +  9^=: -57. 
1_1      1_11      3_3      j4_71      3      4_^_20 

'x      y'^z~6''     X      y'^'6z~18'     x'^y      Yz~  21 

5-  i(i/-^)=i(y--'^0  =  52-4'^.     .?/  +  ^  =  2.r  +  l. 
6.  x  +  2y  +  '6z  +  -iu  =  20,         x  +  2y -\-'6z-4:U  =  12, 

x  +  2y-Zz  +  4:U=:S,  x-2y  +  dz  +  A:U  =  8. 

1.  2x+  y+   2-16, 
x  +  2y+   2  =  9, 
x-\-  y  +  2z  =  3. 

10.  'Ll^  =  z-2, 

70 


8.  cy-^bz  =  a, 

9. 

x  +  y  =  U, 

az  +  ex  =  b, 

x  +  z  =  l6, 

bx  +  ay  =  c. 

y-hz  =  l8. 

9         Q         J. 

11.  -+-  +  -=29, 
X      y      z 

12. 

^+1  =  1, 
a      b        ' 

X     y     z 

^+?=i, 

a     c       ' 

X     y     z 

|+?=i. 

^      c 

^  =:  Z(;  +  8, 

Z 

x-\-y  =  4:2c, 

7U  —  1=  Z. 

13.  2(.r  +  l)-3(?/-l)  +  2-2  =  2,  14.  a;  +  ?/  +  2  =  9, 
2(^  +  1) +4(/y  +  l) -5(2-1)  =  3,  a:  +  2?y  +  42=15, 

3(2a;  +  2)-2(?/-l) +  3(2  +  1)  =  29.  ^  +  3^  +  92  =  23. 

15.  (:.  +  l)(5^-3)  =  (T.T  +  l)(2^-3),  17.  f^+i_f=,^ 
(4:.-l)(2  +  l)  =  (:.  +  l)(22-l),  ^      J      ^^ 

(;y  +  3)(2  +  2)  =  (3?/-6)(32-l).  x-y^z  =  '' 

,,    92537      15,  «^c^ 

16. = =-  +  -  =  4.  +_+_  =  ^. 

X     y      z      X      y     Zz  x     y     z 

18.  a;  +  a2/  +  fl^2  +  A  +  «*  =  0,     x+by-\-b'^z+bht  +  b^  =  0, 
x-^cy  +  c^z+  chi  +  6'*  =  0,     a;  +  dy  +  ^^22  + 1/^//  +  ^*  =  0. 

19.  ?<  + 1'  +  ?^  +  .T  +  ?/  =:  10,  ??  +  ?r  +  a;  +  _?/  +  2  =  15, 
w  +  a;  +  y  +  2  +  ?<  =  ]3,  a;  +  ?/+2  +  ?^+t'  =  ll, 
^+2  +  ?t  +  i;  +  w  =  14,     2  +  ?^  +  2;  +  ?(^  +  2;  =:  12. 


90  SIMPLE   EQUATIONS.  [  ill,  Pr. 

NOT    ALL    THE    UNKNOWN    ELEMENTS    INVOLVED    IN    EVERY 

EQUATION. 

Note  1.  An  unknown  element  that  does  not  appear  in  any 
equation  may  be  considered  as  already  eliminated  from  it, 
and  the  work  is  shortened  by  so  much;  those  unknown  ele- 
ments that  are  in  the  fewest  equations  may  be  eliminated  first. 
E.g.,  to  find  X,  y,  z,  t,  u  from  the  set  of  equations 

^x-2z+u   =41,       (1)         ly-bz-t  =12,        (2) 
4^/-3.^'^-2w=   5,       (3)         ^y-4:U  +  U=^   7,        (4) 
^iz-bu  =  ll'.       (5) 
of  these  equations,  x  appears  in  two,  y  in  three,  z  in  three,  w 

in  four,  t  in  two; 
equations  1,  3  may  be  combined  to  eliminate  a:,  and  equations 
2,  4  to  eliminate  t,  and  there  result  two  new  equations, 
involving  y,  z,  u ; 
these  two  equations  may  be  combined  to  eliminate  ?/,  and 

there  results  one  equation,  involving  z,  u^ 
this  last  equation  may  be  combined  with  equation  5  to  elimi- 
nate either  z  or  u  at  pleasure. 

PARTICULAR   ARTIFICES. 

Note  2.  The  equations  may  have  a  certain  symmetry  as 
to  the  unknown  elements,  or  functions  of  them,  that  permits 
shorter  processes  than  those  of  the  general  rule;  sometimes 
the  sum  of  such  unknown  elements,  or  of  the  functions,  may 
be  got  first. 
E.g.,  to  find  X,  y,  z  from  the  set  of  equations 

1      1_4       1      1^_11      1     1_1_. 

x^y~'\h'    y^z'm''    z~^x~4:' 

222         7        1,11        7  rnvi.o 

then  •.•-+-+-  =  1-^,     -  +  -  +  -=  --,  [add,  div.  by  2. 

X     y     z       10      x     y      z      20 

,i_^_ii_]_  1  ^ 'i.,!  ^  1     L=^_A=l. 

'''x~26     60~6'y~20     4  ~  16'     z       20     15      12' 
.•,x  =  G,  ^  =  10,  ;2  =  12. 


5,  §3]  THREE  OR   MORE   UNKNOWN   ELEMENTS.  91 

QUESTIONS. 

Solve  the  systems  of  equations : 

1.  3x-4i/  +  3z  +  3v-6u  =  ll,         2.   3z  +  8u  =  33, 
3x-Di/  +  2z-4:ic    =11,  7x-2z-\-3u  =17, 

oz -\- 4:11 +  2v-2x    =3,  4:i/-2z  +  v     =11, 

I0i/-3z  +  3u-2v  =  2,  4^-32^  +  2^  =  9, 

6u-3i^  +  4:X-2y  =6.  oi/-3x-2u  =  8. 

3.  x  +  2i/-3z    =-1,        4.  x-{-ij  +  z  =  0, 

4:X-4:i/-z    =8,  {b  +  c)x-\-{c-\-a)2/-\-(a  +  b)z  =  0, 

3x-h8i/  +  2z  =  ~D.  hex -\- cay  +  aiz  =  1, 

5.  5.r-2(^  +  ^  +  f)  =  -1,      -l2y-\-3{z  +  v  +  x)  =  3, 
4:Z-3{v  +  x-^y)=2,         Sv-{x  +  y  +  z)=-2, 

6.  Three  cities,  A,  B,  C,  are  at  the  corners  of  a  triangle; 
from  A  through  B  to  C  is  118  miles;  from  B  through  C  to 

A,  74  miles;  from  C  through  A  to  B,  92  miles:  how  far  apart 
are  the  three  cities  ? 

7.  The  sum  of  tliree  numbers  is  70;  the  second  divided 
by  the  first  gives  2  for  the  quotient  and  1  for  the  remainder, 
and  the  third  divided  by  the  second  gives  3  for  both  quotient 
and  remainder:  find  the  numbers. 

8.  A,  B,  C  are  three  towns  forming  a  triangle;  a  man  has 
to  walk  from  one  to  the  next,  ride  thence  to  the  next,  and 
drive  thence  to  his  starting  point;  he  can  walk,  ride,  and 
drive  a  mile  in  «,  h,  c  minutes  respectively;  if  he  start  from 

B,  he  takes  a  +  c  —  h  hours;  if  from  C,  h  +  a  —  c  hours;  if  from 
A,  c-\-l)  —  a  hours:  find  the  length  of  the  circuit. 

9.  A  number  is  expressed  by  three  figures,  whose  sum  is 
19;  reversing  the  order  of  the  first  two  figures  diminishes 
the  number  by  180,  and  interchatiging  the  last  two  increases 
it  by  18 :  what  is  the  number  ? 

10.  A's  money  in  9  years  at  6  %  will  produce  as  much  in- 
terest as  B's  and  C's  together  in  4  yrs.  6  raos.  at  4  ^;  B's  in 
8  yrs.  at  5  ^  as  much  as  A's  and  C's  in  3  yrs.  4  mos.  at  6  ^; 
C's  in  7  yrs.  at  3  fc  §42  more  than  A^s  and  B's  in  3  yrs.  at  4  ^: 
how  much  money  has  each  man? 


92  SIMPLE  EQUATIONS.  tin,  Pr. 

♦general  forms. 

Note  6.  The  two  equations      ax  +  by  =  c,      a'x-\-l'y  ~  c', 
are  the  type-forms  of  every  pair  of  simple  equations  that  in- 
volve the  same  two  unknown  elements;  their  solution  gives 
X  =  {cb'-c'b)/{aV-a'b\   y  =  {cw' -a'c)/{ab' -a'b\ 

The  values  of  x,  y  for  a  particular  pair  of  equations  depend 
on  the  values  of  a,  b,  c,  «',  b',  c\  and  an  examination  of  the 
possible  values  and  relations  of  these  known  elements  will 
determine  the  possible  roots  of  the  pair  of  equations.  There 
are  three  general  cases: 

(pi)    ab' i^a'b\  then  x,  y  have  single  values,  positive,  negative, 
or  zero,  that  satisfy  both  the  equations. 

(Z>)   ab' =  a'b,     cb' i^c'b\    then    ac'^a'c,    a;  =  co,     y  —  cxi. 

Here  a/a*  —  b/V 4^c/c' \  the  equations  are  inconsistent, 
and  they  can  be  satisfied  by  no  finite  values  of  x,  y. 

The  infinite  values  may  be  interpreted  by  saying  that  if 
a,  a' ,  b,  b' ,  any  of  them,  take  changing  values,  and  if  ab'  ^a'b, 
but  ab'  approach  nearer  and  nearer  to  a'b,  then  x,  y  grow 
larger  and  larger  without  bounds. 

(c)    ab'^a'b,   cb'  =  c'b\   then  ac'  =  a'c,  x  =  0/Oy    y  =  0/0. 

Here  a/a' —  b/b' —  c/c' \  the  equations  differ  by  a  factor 
only,  and  the  values  sought  are  indeterminate. 

The  general  forms  of  simple  equations  involving  three  un- 
known elements  are     ax-^by-\-cz  —  d,    a'x  +  Vy  +  c'z  —  d\ 
a"x  +  b"y  +  c^'z  =  ^",     whose  solution  gives 

_  f^yc^'_j,'^a')^(r(y^c-bc")-hd''{bc/-b'c) 
^~  a{b'c"-b"c')+a\b"c-bc")  +a'\bc'-b'cy 

_  d(a'<r-a''c/)  +  d\a"c-ac")  +  d"{ac/-a'c) 
y~  b{ii'c"-a"c')^b\a"c-ac")^b'\ac'-a'c)  ' 

_  d(a'V'-(t"b')-Vd'{a"b-nh")^d"{aV-a'b) 
^  -  c\a'b"-a"b')  ^c\a"b-ab")  +c"{ab'-a'bY 

and  all  of  these  denominators  have  the  same  value;  but  the 
sign  of  the  second  is  opposite  to  that  of  the  first  and  third. 


5,  §3]  THREE  OR  MORE   UNKNOWN   ELEMENTS.  93 

Various  relations  among  the  coeflficieiits  may  be  considered: 
If  d,  d' ,  d"  be  all  zero,  the  values  of  x,  y,  z  are  zero,  unless 
the  denominator  is  also  zero,  and  then  these  values  are  inde- 
terminate, and  the  given  equations  are  not  all  independent. 

If  dy  d',  d"  be  not  all  zero,  but  the  denominator  be  zero, 
the  equations  are  inconsistent. 

For  if  the  first  equation  be  multiplied  by  h'c"  —  Vc' y  the 
second  by  h"c—lc",  the  third  by  Ic'  —  l'c,  and 
the  results  be  added, 
then  the  coefficients  of  y  and  z  vanish  identically,  and  that  of 
a:  is  a{b'c"-l"c')^a\l"c-hc")^a"{hc'-h'c\  i.e., 
zero,  while  the  second  member  is  not  zero. 

QUESTIONS. 

1.  In  the  pair  of  equations  given  in  note  6,  what  relation 
between  the  products  a'h,  ab'  makes  the  denominator  of  the 
value  of  X  positive?  negative  ?  zero  ? 

So,  what  relation  between  the  products  ac\  a'c  makes  the 
numerator  positive  ?  negative  ?  zero  ? 

If    ch'>c'b    and     ah' <a'h,     is  .r  positive  or  negative? 

2.  If  the  numerator  of  a  fraction  be  0,  and  the  denominator 
not  0,  what  is  the  value  of  the  fraction  ?  if  the  denominator 
be  0  and  the  numerator  not  0?  if  both  be  0? 

3.  If  db'  =  a'h  and  ch'  ^c'h^  find  the  value  of  the  fraction 
h/h',  and  so  show  that  ac'  ^a'c,  and  that  x,  y  are  both  infinite. 

4.  If  ah'  =  a'h  and  ch'  =  c'h,  show  that  ac'  =  a'c. 

What  then  is  the  numerator  of  the  value  of  :?;?  what  the 
denominator  ?  of  the  value  of  y  ? 

Has  the  fraction  0/0  any  definite  value  ? 

5.  Given  the  three  simple  equations 

ax  4-  hy  =  c,     a'x  +  h'y  =  c',     a"x  +  h"y  =  c", 
solve  the  first  two,  substitute  the  values  of  x,  y,  so  found, 
in  the  third,  and  show  that 
a(h'c"-h"c')+a'{h"c-hc")^a"(hc'-h'c)-0, 
is  the  equation  of  condition  for  the  consistency  of  the 
three  given  equations. 


94  SIMPLE  EQUATIONS.  [m, 

§4.   QUESTIONS   FOR  REVIEW. 

Define  and  illustrate: 

1.  An  equation;  an  identity;  an  axiom. 

2.  Known  elements;  unknown  elements;  the  solution  of  an 
equation ;  the  roots  of  an  equation. 

3.  A  simple  equation  involving  one  unknown  element;  two 
unknown  elements;  three  unknown  elements. 

4.  A  pair  of  simultaneous  equations;  elimination. 
State  the  axiom: 

5.  Of  equality;  of  addition;  of  subtraction;  of  multiplica- 
tion; of  division;  of  involution;  of  evolution. 

Give  the  general  rule,  with  reasons  and  illustrations,  for: 

6.  Solving  a  simple  equation  with  one  unknown  element. 

7.  Eliminating  an  unknown  element  from  a  pair  of  simul- 
taneous simple  equations  by  addition  and  subtraction;  by 
comparison;  by  substitution. 

8.  Solving  a  pair  of  simultaneous  simple  equations. 

9.  Solving  a  set  of  simple  equations  involving  three  or 
more  unknown  elements. 


Exhibit  the  general  forms,  and  explain  the  special  cases,  for: 

10.  A  simple  equation  involving  one  unknown  element. 

11.  A  pair  of  simple  equations  involving  the  same  two  un- 
known elements. 

What  difficulty  arises: 

12.  If  one  of  the  two  equations  be  dependent  on  the  other? 

13.  If  there  be  more  conditions  than  unknown  elenients? 

14.  If  there  be  fewer  conditions  than  unknown  elements? 


Explain  what  is  meant: 

15.  By   putting  a   problem    into   equation;   by  solving   a 
problem;  by  checking  the  work. 

16.  By  the  discussion  of  a  problem  whose  answer  is  numer- 
ical; of  a  problem  whose  answer  is  literal. 


§4]  QUESTIONS    FOR  REVIEW.  Uo 

IT.  The  sum  of  the  three  figures  of  a  number  is  9;  the  first 
figure  is  an  eighth  of  the  number  made  up  of  the  last  two 
figures  taken  in  order,  and  the  last  figure  is  an  eighth  of  the 
number  made  up  of  the  first  two  tigures:  find  the  number. 

18.  A  boatman  rows  down  the  river  42  miles  in  3  hours; 
returning,  he  finds  the  current  only  two  thirds  as  strong,  and 
it  takes  him  10^  hours:  find  how  fast  he  can  row  in  still  water, 
and  how  fast  the  river  ran  at  first. 

19.  At  3^  miles  an  hour,  I  can  walk  from  p  to  Q  in  a  certain 
time;  but  at  the  rate  of  3  miles  going  and  4  miles  returning, 
it  takes  me  five  minutes  longer:  how  long  is  the  round  trip  'i 

20.  A  crew  that  can  pull  9  miles  an  hour  in  still  water  takes 
twice  as  long  to  come  up  a  river  as  to  go  down  it:  find  the 
velocity  of  the  current. 

21.  If  a  rectangle  be  made  3  feet  longer  and  3  feet  broader, 
the  area  is  102  square  feet  greater;  but  if  it  be  shortened  5 
feet,  and  widened  1  foot,  the  area  is  16  square  feet  less:  find 
the  length  and  breadth. 

22.  If  a  concert  room  contained  10  more  benches,  one  per- 
son less  might  sit  on  a  bench;  if  it  contained  15  fewer,  2  more 
persons  must  sit  on  a  bench:  how  many  benches  are  there 
and  how  many  people  on  a  bench  ? 

23.  The  perimeter  of  a  rectangular  field  is  TO  rods;  if  the 
width  were  increased  one  rod  and  the  length  diminished  two 
rods,  the  width  would  be  eight  ninths  of  the  length:  find  the 
area  of  the  field. 

24.  The  contents  of  a  vessel  is  60  per  cent  alcohol  and  the 
rest  water;  after  drawing  out  10  gallons  of  the  mixture  and 
filling  up  the  vessel  with  water,  42 f  per  cent  of  the  contents 
is  alcohol:  find  the  capacity  of  the  vessel. 

25.  A  cask  contains  18  gallons  of  wine  and  12  of  water,  and 
another  contains  3  gallons  of  wine  and  9  of  water:  how  many 
gallons  must  be  drawn  from  each  cask  to  make  the  mixture 
contain  7  gallons  of  wine  and  7  of  water  ? 


96  SIMPLE   EQUATIONS.  [in, 

26.  Every  year  a  merchant  adds  40  per  cent  to  his  capital, 
but  takes  out  $3000  for  expenses;  at  the  end  of  the  third  year, 
after  deducting  his  13000,  he  finds  that  he  has  doubled  his 
original  capital  and  has  11800  besides:  how  much  had  he  at 
first  ?  how  much  at  the  end  of  each  year  ? 

27.  A  lady  receives  $21G0  yearly  interest  on  her  capital,  but 
if  it  were  loaned  at  a  half  of  one  per  cent  higher,  she  would 
receive  $240  more:  find  her  capital  and  the  rate  of  interest. 

28.  How  much  pure  copper  must  be  added  to  35  pounds  of 
silver,  15  parts  pure  out  of  16,  so  that  the  mixture  shall  con- 
tain 4  parts  of  pure  silver  to  one  part  of  alloy  ? 

29.  In  a  naval  action,  a  third  of  a  fleet  was  taken,  a  sixth 
was  sunk  and  two  ships  were  burnt;  afterwards  a  seventh  of 
the  remaining  ships  were  lost  in  a  storm,  and  only  24  ships 
were  left:  how  large  was  the  fleet  at  first? 

30.  A  and  B  begin  a  game,  both  having  the  same  sum  of 
money,  and  the  loser  is  always  to  pay  the  winner  a  dollar  more 
than  half  what  the  loser  has;  after  B  has  lost  one  game  and 
won  one,  he  has  twice  as  much  money  as  A:  how  much  had 
he  at  first  ? 

31.  A  besieged  garrison  had  bread  enough  to  last  6  weeks, 
giving  each  man  10  oz.  a  day;  but,  after  they  lost  1200  men 
in  a  sally,  the  allowance  was  increased  to  12  oz.  a  day,  and 
the  bread  lasted  8  weeks:  how  many  men  were  there  at  first  ? 

32.  A  man  and  his  family  use  a  barrel  of  flour  in  m.  days, 
but  when  the  man  is  away  it  lasts  n  days  longer:  how  long 
would  it  last  the  man  alone  ? 

33.  A  man  was  engaged  to  work  48  days  at  two  dollars  a 
day  and  his  board,  which  was  estimated  at  a  dollar  a  day;  at 
the  end  of  the  time  he  received  $42,  his  employer  having  de- 
ducted the  cost  of  board  for  the  days  he  was  idle:  how  many 
days  had  he  worked  ? 

34.  Out  of  a  certain  sum  of  money,  a  man  paid  his  creditors 
$432,  lent  a  friend  a  third  of  the  remainder,  and  spent  a 
quarter  of  what  was  then  left;  after  this  he  had  a  fifth  of  the 
original  sum:  how  much  money  had  he  at  first? 


M]  QUESTIONS  FOR  REVIEW.  97 

PROBLEMS    OF    PURSUIT. 

35.  A  is  12  miles  behind  B,  and  gains  2  miles  an  hour;  when 
will  they  be  together?  When,  if  A  have  7n  miles  to  gain,  and 
gain  h  miles  in  k  hours  ? 

36.  A  is  180  miles  east  of  B,  A's  train  runs  25  miles  an  hour 
and  B's  20  miles:  when  will  they  be  cogether  if  A  goes  west 
and  B  east?  A  east  and  B  west?  both  west?  both  east?  wlien, 
if  A  be  ?/^  miles  east  of  B  and  they  travel  at  a,  b  miles  an  hour  ? 

Interpret  the  results  if  tii  be  positive,  negative;  a,  positive, 
negative;  J,  positive,  negative;  a>b'y  a  =  lj;  a<b. 

37.  B  has  a  start  of  h  hours,  A  goes  c  miles  an  hour  faster 
than  B,  and  overtakes  him  in  h  hours:  what  is  each  man's  rate  ? 

Interpret  the  results  if  c  be  negative;  h  negative;  k  negative. 

38.  A,  B  starting  together  walk  around  a  track;  at  the  end 
of  half  an  hour  A  has  made  3  circuits  and  B  ^\:  when  will  B 
next  pass  A?  when,  if  A  has  made  a  circuits  and  B,  b  circuits? 

39.  The  circle  of  a  clock  face  is  divided  into  12  hour-arcs: 
how  many  arcs  does  the  hour  hand  pass  over  in  an  hour  ?  in 
two  hours  ?  •  •  •  in  twelve  hours  ?  the  minute  hand  ? 

How  many  arcs  does  the  minute  hand  gain,  over  the  hour 
hand,  in  one  hour  ?  in  two  hours?  •  •  •  in  twelve  hours  ? 

40.  How  many  hour-arcs  has  the  minute  hand  gained,  over 
the  hour  hand,  when  they  are  together  between  one  and  two  ? 
between  two  and  three  ?  •  •  •  between  ten  and  eleven  ? 

41.  At  what  time  are  the  hour  hand  and  minute  hand  op- 
posite to  each  other  between  twelve  and  one?  between  one 
and  two  ?  •  •  •  between  five  and  seven  ?  •  •  •  between  eleven  and 
twelve  ?  At  what  times  between  twelve  and  twelve  are  the  two 
hands  at  right  angles  to  each  other  ? 

42.  If  A,  B,  C  starting  together  at  noon  walk  around  a 
track,  A  ten  times  in  an  hour,  B  twelve  times,  and  C  fifteen 
times,  when  will  A  and  B  be  together  for  the  first  time  ?  B 
and  C?  C  and  A?     When  will  they  be  all  together? 

When  will  A  be  midway  between  B  and  C?  B  midway  be- 
tween C  and  A  ?  C  midway  between  A  and  B  ? 


98  SIMPLE  EQUATIONS.  [m, 

43.  If  the  hour,  minute,  and  second  hands  of  a  clock  all  turn 
upon  the  same  ceutre,  at  what  times  will  the  minute  and 
second  hands  be  together  ?  at  what  times  will  each  of  the  three 
hands  be  midway  between  the  other  two  ? 

44.  A  passenger  train  200  ft.  long  passes  a  freight  train 
680  ft.  long  in  30  seconds  when  they  are  running  in  opposite 
directions,  and  in  one  minute  when  in  the  same  direction:  find 
the  rate  of  each  train. 

PERCENTAGE   AND   SIMPLE  INTEREST. 

45.  Put  h  for  the  basis  of  percentage,  j)  for  the  percentage, 
r  for  the  rate,  a  for  the  amount  of  the  basis  and  the  percent- 
age, V  for  the  net  value  of  the  basis  less  the  percentage;  then, 
by  definition,         r—p/h,        a  —  h+j),        v  —  b—p. 

From  these  three  fundamental  equations  show  that: 
b=p/r,    p  =  b'r,    a  =  b'{l-\-r),     b  =  a/(l  +  r), 
v  =  b{l-r),       b  =  v/{l-r). 
Translate  these  six  formulae  into  theorems  and  into  rules. 

46.  Put  p  for  the  principal,  r  for  the  yearly  rate,  t  for  the 
time  in  years,  i  for  the  simple  interest,  a  for  the  amount;  then, 
by  definition,     i=p-r- 1,     a—p-\- i. 

From  these  two  fundamental  equations,  show  that 

p  =  i/ri,  r  =  i/ptf  t  —  i/pv,  a—p'{\-\-  rt),  p  —  a /{I  +  W). 
Translate  these  five  formulse  into  theorems  and  into  rules. 

47.  Bank  discount  is  simple  interest  prepaid,  and  the  bank 
present  ivorth,  or  proceeds ,  is  the  principal  less  the  discount. 

Put  V  for  the  present  worth,  then,  by  definition,    v—p  —  i. 

Show  that     v=p'{l~rt),    p  =  v/(l—rt). 

Translate  these  two  formulae  into  theorems  and  into  rules. 

Show  the  relation  between  the  true  present  worth  and  the 
bank  present  worth  for  ji  given  principal,  rate,  and  time;  and 
that  between  the  true  rate  earned  and  the  bank  rate. 

AVERAGES. 

48.  If  the  foreman  gets  $5  a  day,  two  sub-foremen  $S  each, 
and  forty  men  12  each,  wliat  is  the  average  daily  wages  for  the 
whole  party  ? 

Replace  the  numerals  by  letters  and  make  a  general  formula. 


§3)  QUESTIONS   FOR  REVIEW.  99 

49.  A  grocer  mixes  20  lbs.  of  tea  worth  50  cents  a  pound 
with  30  lbs.  worth  40  cents,  and  50  lbs.  worth  30  cents,  and 
sells  the  whole  at  45  cents;  what  is  the  real  value  ?  the  profit 
on  one  pound  ?  the  rate  of  profit  ?     Make  a  general  formula. 

50.  With  debts ^J*!,  2^-2,  Jh,  •  •  -due  at  times  t^,  t^,  t^-  •  -from 
a  fixed  date,  find  a  siugle  date  when  the  whole  may  be  paid 
witliout  loss  to  debtor  or  creditor. 

Discuss  the  problem  if  some  of  the  /'s  or^j's  be  negative. 

WOKK. 

51.  A  man  can  do  a  piece  of  work  in  w  clays:  what  part  of 
it  can  he  do  in  one  day  ?  in  two  days  ?  in  three  days  ?  in  ten 
days?  in  7i  days?  in  2n  days  ? 

52.  A  man  can  do  n  units  of  work  in  one  day:  in  what  part 
of  a  day  can  he  do  one  unit?  two  units?  three  units?  ten 
units?  n  units?  2)i  units? 

53.  A  can  do  a  piece  of  work  in  a  days,  B  in  Z*  days,  C  in 
c  days:  what  part  of  the  work  can  A  and  B  together  do  in 
one  day  ?     B  and  C  ?     C  and  A  ?     A,  B  and  C  ? 

54.  A  can  do  a  units  of  work  in  a  day,  B,  I)  units,  C,  c  units: 
in  what  part  of  a  day  can  A  and  B  together  do  a  unit  of  work? 
B  and  C?     C  and  A  ?     A,  B  and  C? 

55.  A  can  do  a  units  of  work  in  ft'  days,  B,  b  units  in 
y  days,  C,  c  units  in  c'  days:  in  how  many  days  can  they  to- 
gether do  a  +  b  +  c  units? 

56.  To  do  a  certain  piece  of  work  A  needs  7n  times  as  long 
as  B  and  C  together;  B,  u  times  as  long  as  0  and  A;  C,  p 
times  as  long  as  A  and  B :  what  relation  have  m,  n,  ])  ? 

57.  A  reservoir  is  filled  by  pipes  A,  B  in  c  hours,  by  pipes 
B,  C  in  ft  hours,  by  pipes  C,  A  in  Z*  hours:  in  what  time  is  it 
filled  by  each  pipe  running  alone  ?  by  the  three  pipes  together? 

58.  A  reservoir  holding  m  gallons  is  filled  by  two  pipes, 
running  a,  b  gallons  an  hour,  and  emptied  by  two  pipes  run- 
ning c,  d  gallons  an  hour:  what  is  the  relation  between  ft,  Z», 
c,  d  so  that,  with  all  the  pipes  running,  the  reservoir,  if  empty, 
shall  be  filled  in  li  hours  ?  if  full,  emptied  in  li  hours? 


100  MEASURES  AND  MULTIPLES.  IIV.Th. 

IV.  MEASURES  AND  MULTIPLES. 


§1.    INTEGERS. 

The  product  of  two  integers  is  a  itiuUi'jjIe  of  either  integer; 
and  either  integer  is  a  measure  of  the  product. 
E.g.,  "^15,  "15,    are  multiples  of     +5,  "5,  "*"3,  "3; 
and     +5,  ~o,  ^S,  "3,     are  measures  of  +15,  and  of  ~15, 

Every  integer  is  a  multiple  of  itself,  its  opposite,  and  *1. 

COMMOJ^"   MULTIPLES  AND   MEASURES. 

If  the  same  number  be  a  multiple  of  two  or  more  integers, 
it  is  a  common  m^iUipIe  of  them;  and  a  measure  of  two  or 
more  integei-s  is  a  common  measure  of  them. 
E.g.,     "^30,  ~30,    are  common  multijiles  of  1,  ~5,  10,  "15,  30; 
and       """S,  ~3,     are  common  measures  of     3,  ~6,     9,  "15,  30. 

The  smallest  of  all  the  common  multiples  of  two  or  more 
integers  is  their  lowest  comiuoii  multiple;  and  the  largest  of 
all  their  common  measures  is  their  liigliest  common  measure. 
E.g.,  *30  is  the  lowest  common  multiple  of     3,  10,  "15,  but 

not  of     3,  "15; 
and  *3  is  the  highest  common  measure  of   G,  "9,  12,  but  not 

of    6,  12. 

Ax.  1.  The  siwif  the  differ eiice,  and  the  j^roduct  of  two  in- 
tegers are  integers. 

Theor.  1.  A  common  measure  of  two  or  more  integers  is  a 
measure  of  their  sum. 

Let  A,  B,  •  •  •  be  any  integers  and  m  a  common  measure  of  them; 
then  is  M  a  measure  of  the  sum  A  +  B+  •  •  • 
For  •.*  A  =  M •  r?,  B  =  M •  ^;, •  •  • ,  and  a,  b,'  -  *  are  integers,      [hyp. 

.'.  A  +  B  +  •  •  •  =  M  •  rt  4-  M  •  Z*  +  •  •  • 

=  M.(^/.  +  /;+...);  [I,  th.  7. 

and  *.*  the  sum  a  +  h+  -  -  '  in  an  integer,  [ax.  1. 

.'.  M  is  a  measure  of  A  +  b  +  •  •  •  Q.E.D.         [df.  msr. 


1,§1]  INTEGERS.  '  ''  "''101 

Cor.  a  common  measure  of  tic o  integers  is  a  measure  of  the 
sum  and  of  the  difference  of  any  7?iuUiples  of  them, 

QUESTIOXS. 

1.  Explain  the  difference  between  a  product  and  a  multiple; 
between  a  divisor  and  a  measure. 

2.  Xame  some  multiples  of  6,  of  "3,  of  0,  of     1. 

3.  Xame  some  measures  of  24,  of  60,  of  ~64,  of  1,  of  0. 

4.  Name  some  common  multiples  of  2,  3,  ~4,  1;  of  5,  7, 11. 

5.  Name  some  common  measures  of  24,  64,  120. 

6.  Name  the  low^est  common  multiple  of  6,  15,  10:  what  is 
their  highest  common  multiple  ? 

7.  Name  the  highest  common  measure  of  "120,  "45,  "60: 
what  is  their  lowest  common  measure?  What  common 
measure  of  two  of  these  numbers  is  not  a  measure  of  the  third  ? 

8.  In  the  proof  of  theor.  1,  why  must  a,  b,-  •  -he  integers  ? 
Explain  the  dependence  of  the  last  statement  of  the  proof 

on  that  just  before  it, 

9.  Prove  that  a  common  measure  of  two  integers  is  a  meas- 
ure of  their  difference  and  of  their  product. 

10.  Prove  that  a  measure  of  the  sum  of  two  or  more  integers, 
if  it  measure  all  of  them  but  one,  measures  that  one  also. 

11.  Prove  that  a  measure  of  an  integer  is  a  measure  of  any 
multiple  of  that  integer;  that  a  multiple  of  an  integer  is  a 
multiple  of  all  measures  of  that  integer. 

12.  What  two  common  measures  have  all  integers? 
"What  other  two  measures  has  any  given  integer  ? 

13.  Why  is  any  positive  integer  a  multiple  of  its  opposite? 
Why  is  any  measure  of  a  negative  integer  a  measure  of  the 

same  integer  with  the  positive  sign  ? 

14.  In  finding  the  highest  common  measure  of  integers, 
will  the  answer  be  affected  if  all  the  integers  be  taken  ^lositive? 

15.  Prove  that  the  sum,  the  difference,  and  the  product  cf 
two  even  numbers  are  even. 


102  MEA-SUKES  AND  MULTIPLES.  [IV,Th. 

Euclid's  process  for  FiNDr^G  the  highest  commoi^" 

MEASURE. 

Theor.  2.  If  the  larger  of  ttvo  integers  he  divided  hy  the 
smaller,  the  common  measures  of  the  divisor  and  the  remai7ider 
are  the  common  measiires  of  the  two  integers. 

If  the  smaller  integer  he  divided  hy  the  remainder,  thih 
divisor  hy  the  second  remainder,  and  so  on,  then  some  re- 
mainder is  zero. 

The  last  divisor  is  the  highest  common  measure  of  the  two 
integers. 
Let  A,  B  be  two  integers,  Qi  the  quotient  of  A  by  b,  and  Ri, 

R2 ,  •  •  •  Rn  the  successive  remainders; 
then  •.•  Ri  =  A  -  B  •  Qi ,  [df.  rem. 

/.  the  common  measures  of  A,  B  are  measures  of  Rj . 

[th.  1,  cr. 
i.e.,       the  common  measures  of  A,  B  are  common  measures  of 

B,  Ri. 
So,     VA  =  B-Qi  +  Ri, 

.'.  the  common  measures  of  B,  Ri  are  measures  of  A, 
i.e.,       the  common  measures  of  B,  Rj  are  common  measures 
of  A,  B ; 

.*.  the  common  measures  of  A,  B  are  the  common  meas- 
ures of  B,  Ri,  and  there  are  no  others.       q.e.d. 

2.  *.•  Ri,  R2-  •  -are  all  integers,  and  grow  smaller  and  smaller, 
/.  some  one  of  them,  say  R^,  is  0.  q.e.d. 

3.  •/  the  common  measures  of  b,  Rj  are  the  common  meas- 

ures of  Ri,R2,  [above. 

/.  the  common  measures  of  A,  B  are  the  common  meas- 
ures of  Ri ,  R2 ;  and  so  on; 

.*.  the  common  measures  of  A,  b  are  the  common  meas- 
ures of  Rn-2^  Rn-i; 
and    *.*  Rn-i  i«  the  highest  common  measure  of  Rn-?,  Rn^i> 

/.  R^_i  is  the  highest  common  measure  of  A,  b.       q.e.d 


2§n  INTEGERS.  103 

If  Hn-i  be  1,  then  a,  b  are  iisiiiilly  said  to  have  no  common 
measure;  for  unity,  being  a  measure  of  all  integers,  is  not  a 
characteristic  common  measui-e  of  A,  B. 

Cor.  Fvcri/  remainder,  Rj,  e^  •  •  •  e„  is  the  difference  of  tivo 
multiples  of  a,  b. 

E.g.,     Ri  =  A-B-Qi, 

R2=:B-Ri.Q2  =  B-(A-B-Qi)-Q2=-A-Q2+B.(1  +  Qi.Q3), 
R3=Rl-R2.Q3=   A-(1  +  Q2-Q3)-B-(Qi4-Q3  4-Q,.Q2.Q3). 

QUESTIONS. 

1.  In  division,  how  does  the  remainder  compare,  in  size, 
with  the  divisor  ?  if  the  divisor  be  divided  by  the  remainder, 
how  does  the  second  remainder  compare  with  the  first  ? 

In  Euclid's  process  how  do  the  remainders  change  ? 
Can  any  remainder  be  a  fraction  or  a  negative  number? 
Why  must  an  exact  divisor  be  reached  at  last  ? 

2.  If  R^  be  the  last  remainder,  for  what  two  things  does 
En-i  stand?  for  what  three  things  does  R„_2  stand  ? 

3.  Show  that  the  following  statements  are  true: 

any  measure  of  B  and  Rj  is  a  measure  of  A, 
any  measure  of  Ej  and  Eg  is  a  measure  of  b, 

any  measure  of  E„_2  and  E^-i  is  a  measure  of  E„_3, 
any  measure  of  En-i  and  E„     is  a  measure  of  R„_2. 

4.  What  measure  of  E„_2  is  found  by  Euclid's  process  ? 
Is  this  measure  also  a  measure  of  E„_i  ? 

Of  what  other  successive  pairs  of  numbers  is  ^n-i  a  measure  ? 

5.  What  is  the  highest  measure  of  e„_i  ?  the  highest  com- 
mon measure  of  E„_i,  En- 2?  of  e^-q,  Rn-3  ?   •  •  •  of  A,  b  ? 

6.  Find  the  highest  common  measure  of  14637,  51306. 

7.  Draw  any  two  unequal  lines  and  call  them  a,  h]  lay  off  I 
on  a  as  many  times  as  it  can  be  repeated;  lay  off  the  re- 
mainder, c,  on  h',  layoff  the  remainder,  d,  on  c,  and  so  on,  till 
there  is  no  remainder:  what  is  true  of  the  last  line  used  ? 

8.  Prove  that  every  common  measure  of  two  integers  is 
the  difference  of  two  multiples  of  the  integers. 


104  MEASURES   AND  MULTIPLES.  [IV,Ths. 

PRIME   ISrUMBERS. 

An  integer  that  has  no  measures  but  itself  and  unity  is  a 
pri))ie   number;    and   two   integers   that   have   no   common 
measure  except  unity  are  prime  to  each  other. 
E.g.,  2,  -3,  5,  ~ 7,  11  are  prime  numbers;  and  9,  "25  are  prime 
to  each  other,  though  not  prime  numbers. 

An  integer  that  can  be  measured  by  another  integer  is  a 
composite  number. 
E.g.,     15,  "35,     whose  prime  measures  are     3,  5;     5,  7. 

Theor.  3.  If  tivo  integers  be  prime  to  each  other,  then  ttvo 
multiples  of  them  can  be  found  such  that  their  difference  is 
unity;  and  conversely. 
For,  let  A,  B  be  two  integers  that  are  prime  to  each  other,  and 

Ri,  R2,  •  •  •  ,  Rn  the  remainders  as  in  theor.  2; 
then  •.*  A,  B  are  prime  to  each  other,  [hyP* 

.%  Rn-i^  their  highest  common  measure,  is  1; 
and   *.•  Rn-i  is  the  difference  of  two  multiples  of  A,  B,  say 
m 'A,  71' B,  [th.  2  cr. 

/.  m-A  — ?i-B  =  l.  Q.E.D. 

Conversely,  let  m-K  —  n-B^l] 
then  •.*  every  common  measure  of  A,  B  is  a  measure  of  1, 
.*.  A,  B  are  prime  to  each  other.  Q.E.D. 

Theor.  4.  If  an  intefjer  be  prime  to  tioo  or  more  integers,  it 
is  prime  to  their  producf. 
Let  A,  B,  c,  •  •  «  be  any  integers  and  p  an  integer  that  is  prime 

to  each  of  tliein; 
tlien  is  p  prime  to  tlie  product  a-b-c-  •  • 
For  take     m,  n;    p,  q;     r,  s;-  -  •  integers  such  that 

m'P  —  n'A  =  i,  ])-r-  rj'ii=l,  r-i^  —  s-c  =  l,'  '  ',  [th.  3. 
then  •.'  {)n  •  p  —  w  •  a)  x  ( ;;  •  p  -  ^y  •  ij)  x  (y  •  p  —  6^  •  c)  •  •  •  =  1, 
i.e.,       h'P-{-k'\'Ti-C' '  '  =1,  wlicreiu  //, -p  is  the  sum  of  all 
tlie  terms  tliat  coutjiiu  v,  and  k=  ±n'q-S'  •  • 
/.  P  is  prime  to  the  product  A  •  B  •  c  •  •  •    q.e.d.   [th.  3  conv. 


3,4,  §1]  INTEGERS.  105 

Cor.  1.  If  ttco  integers  he  prime  to  each  other,  so  are  their 
positive  integer  powers, 

QUESTION'S. 

1.  !N^ame  seven  prime  integers;  the  only  even  prime  integer. 

2.  If  all  the  integers  from  1  to  1000  be  written  in  order, 
and  all  the  even  numbers  be  struck  out,  every  third  number 
from  3,  every  fifth  from  5,  and  so  on,  what  numbers  remain  ? 

3.  Name  five  integers,  no  one  of  which  is  prime,  but  which 
are  all  prime  to  each  other. 

4.  Is  it  possible  for  an  integer  to  have  but  one  factor  ? 
Separate  36  into  sets  of  two  factors,  of  which  one  shall  be 

2,  3,  4,  6,  9,  12,  18,  36,     in  turn :  were  all  the  possible  sets 
obtained  before  the  entire  series  of  divisors  were  tried  ? 

In  trying  all  possible  factors  of  an  integer  in  the  order  of 
their  size,  how  far  need  the  process  be  carried  ? 

5.  If  two  integers  that  are  prime  to  each  other  be  subjected 
to  Euclid's  process,  what  is  the  last  remainder?  the  last 
divisor  ?  the  last  remainder  but  one  ? 

6.  Find  two  multiples  of  9,  17,  whose  difference  is  1;  so,  of 
5,  13;  of  11,  13;  of  13,  17;  of  17,  19;  of  13,  19. 

7.  What  is  the  converse  of  a  theorem  ? 

State  the  converse  of  theor.  3  as  a  separate  theorem. 

8.  In  the  converse  of  theor.  3,  why  have  A,  B  no  common 
measure  but  1  ? 

9.  What  statement  in  theor.  4  is  a  direct  application  of 
theor.  3  ?  what,  of  the  converse  of  that  theorem  ? 

10.  Are  all  prime  numbers  prime  to  each  other? 

If  an  integer  be  resolved  into  its  prime  factors,  what  rela- 
tion have  other  prime  numbers  to  tliese  factors  separately? 
What  relation  have  these  numbers  to  the  original  number  ? 

11.  Prove  that  an  even  number  can  not  measure  an  odd 
number;  that  a  number  liaving  any  even  factor  is  even;  and 
that  the  product  of  any  number  of  odd  factors  is  an  odd 
number.    Is  the  sum  of  two  odd  numbers  odd  or  even  ? 


106  MEASURES  AND  MULTIPLES.  tlV,THs. 

Cor.  2.  If  an  integer  measure  the  2)rocluct  of  tioo  integers 
and  he  j)rin}e  to  one  of  them,  it  measures  the  other. 
Let  A,  B  be  two  integers  and  P  an  integer  that  measures  the 

product  A-B  and  is  jorime  to  a;  then  p  measures  b. 
For,  let  m-A,  u-^he,  two  multiples  of  a,  p  such  that 

wA-wp  =  l;  [th.  3. 

then     {m  •  a  —  7^  •  p)  •  b  =  '/?i  •  A »  b  —  ?i  -  p  •  b  =:  b, 
and  \'  P  measures  both  a-b,  and  p-b,  H^yPv  df-  nisr. 

.'.  p  measures  B.  Q.e.d.         [th.  1  cr. 

Cor.  3.  If  a  priine  numher  7neasure  the  product  of  two  or 
more  integers,  it  ^neasures  at  least  07ie  of  them. 

Cor.  4.  If  a  prime  miniber  measure  a  positive  integer  poiuer 
of  an  integer,  it  measures  the  integer;  and  if  the  integer,  then 
the  power. 

Cor.  5.  If  an  integer  he  measured  hy  two  integers  that  are 
prime  to  each  other,  it  is  7neasured  hy  their  p>ro duct. 
For,  let  A  be  an  integer  and  P,  Q  integers  prime  to  each  other 

that  measure  a;  and  let  a=:B-p; 
then  •.•  Q  measures  the  product  b-p  and  is  prime  to  p,     [hyp. 
/.  Q  measures  b,  [cr.  2. 

i.e.,       ^;i.Q  =  B,     wherein  7?i  is  some  integer;  [df.  msr. 

.'.  m-P-Q  =  B-P,     and   the   product   p-q   measures    the 
product  B  •  P, 
i.e.,       the  product  p»q  measures  A.  q.e.d. 

Theor.  5.  An  integer  can  he  resolved  into  prime  factors  in 
hut  one  way. 
Let  the  integer  N"  be  the  product  a'^-b^-c"  •  •  •  wherein  A,  b,  c, 

•  •  •  are  primes,  and  a,  h,  c,  '  *  •  are  positive  integers; 
then  K  has  no  other  prime  factor. 

For  *.•  any  other  prime,  F,  is  prime  to  each  of  the  prime  factors 

A,  B,  c,  •••,  [hyp. 

and  to  their  powers  A",  b'',  C,  •  •  •,  [th.  4  cr.  1. 

.*.  F  is  prime  to  the  product  N.  q.e.d.         [th.  4. 


4,5,§1]  INTEGERS.  107 

And  A  can  be  a  factor  of  x  not  more  than  a  times,  b  not  more 
than  h  times,  and  so  on. 

For  •.•x  =  A^.B*'-c^- •  •, 

.-.  x/a"  =  b^.c''.  • . ; 
and  •.*  A  is  prime  to  b^-c'"'  •  •,  [th.  4. 

.'.  A  is  a  factor  of  x  not  more  than  a  times. 
So,  A  can  be  a  factor  of  x  not  fewer  than  a  times;  and  so  with 
b,  with  c,  and  with  the  other  prime  factors.        q.e.d. 

Cor.  a  coiumon  measure  of  two  or  more  integers  can  con- 
tain no  factor  that  is  not  in  all  of  them, 

QUESTIOX^S. 

1.  In  the  proof  of  theor.  4  cor.  2,  are  m,  n  given  numbers  ? 
Are  they  numbers  assumed  at  random  ? 

Why  does  p  measure  ??iAB  ?  why  does  p  measure  b  ? 

2.  So,  if  P  measure  ab,  what  kind  of  number  is  ab/p  ? 
If  P  be  prime  to  a,  is  a/p  an  integei-  ?  what  then  is  b/p  ? 

3.  If  an  even  number  be  measured  by  an  odd  number,  tlic 
quotient  is  even. 

4.  In  the  proof  of  cor.  3,  if  p  measure  the  product  a-b-c-d 
•  •  •  and  be  prime  to  a,  of  what  is  p  a  measure  ? 

So,  if  P  be  prime  to  a,  b,  c,  d,  of  what  is  p  a  measure  ? 

5.  Cor.  4  is  found  by  inserting  the  word  equal  in  cor.  3. 

6.  A  composite  number  may  measure  the  product  of  two  or 
more  integers  and  not  measure  either  of  them  separately. 

7.  If  A  be  measured  by  the  integers  p,  q,  e,  prime  to  each 
other,  and  if  a  =  ???-p-q,  then  r  measures  m,  and  the  product 
p-Q-R  measures  A. 

What  application  is  here  made  of  theor.  4  ? 

8.  In  the  proof  of  theor.  5,  if  p  be  prime  to  a,  why  is  it 
prime  to  A"  ?  why  prime  to  x  ?  If  the  factors  of  X"  be  all 
different,  wdiat  are  tlie  values  of  a,  b,-  -  -  ? 

9.  If  two  integers  have  several  common  prime  factors,  tlie 
product  of  these  factors  is  a  common  measure  of  the  integers. 

Wliat  factors  are  found  in  their  highest  common  measure  ? 


108  MEASURES  AND  MULTIPLES.  [IVc 


§2.   ENTIRE   FUNCTIONS   OF   ONE   LETTER. 

An  algebraic  expression  whose  value  depends  on  the  value 
of  a  single  letter  is  a  function  of  that  letter;  the  letter  is  the 
letter  of  arrangement. 

E.g.,  the  value  of  the  expression     a^  —  ^x^  +  ^x  +  b     depends 
on  the  value  of  x,  and  it  is  known  when  x  is  known. 
If  the  value  of  an  expression  depend  on  the  values  of  two 
or  more  letters  the  expression  is  a  function  of  those  letters. 
E.g.,  the  value  of    a^  —  2x^y-'^  +  'Sxi/-^-\-5y~^    is  known  when 
the  values  of  x,  y  are  known. 
The  letter/ stands  for  the  word  function. 
'Ei.g.ffx  means  a  function  of  x,  and  fa  is  what  fx  becomes 
when  X  is  replaced  by  a. 
If  there  be  more  than  one  function  of  x,  such  functions 
may  be  distinguished  as /^j,  ^x,  cpx,  -  •  •  or  f^x,f^x,f^x,-  •  • 

An  entire  function  of  one  letter  is  the  sum  of  positive  inte- 
ger powers  of  that  letter,  with  or  without  coefficients. 
E.g.,     a:^ 4- 3.^•^  +  3a:  + 1     is  an  entire  function  of  x  of  the  third 

degree,  whose  coefficients  are  integers. 
So,     iy^  —  y^  +  y  —  i     is  an  entire  function  of  y. 

So,  if  n  be  a  positive  integer,     ax''-hbx''-^-\-cx''-^ hicx  +  l 

is  an  entire  function  of  x  of  the  7iih.  degree,  with  literal 
coefficients,  which  may  be  either  integers  or  fractions. 

In  general,  the  definitions  and  principles  established  for 
integers  apply  with  slight  changes  to  entire  functions  of  one 
letter,  and,  for  the  most  part,  they  are  here  stated,  proved, 
and  illustrated  in  the  same  words. 

The  product  of  an  entire  function  of  one  letter  by  another 
such  function  is  a  mi(lli})le  of  either  function,  as  to  that  letter, 
and  either  function  is  a  nwasure  of  the  product. 
E.g.,    «*  — 3rt^  +  3«  — 1     is  a  multiple  of     a^  —  2a-\-\,     a  —  \, 
and      a^-'ia-^-l,   a  — I     arc  measures  of    a^  —  da^  +  3«  —  1. 


§2]  ENTIRE   FUNCTIONS   OF   ONE   LETTER.  109 

So,     S{:c  —  a){y  —  h)     is  a  measure  of     2m (.6^  —  a^){y  +  h)     as  to 

r/,  m,  X, 
and  a  multiple  of  it  as  to  the  numerals; 
but  neither  measure  nor  multiple  as  to  y,  nor  as  to  J. 

Every  entire  function  of  one  letter  is  both  a  multiple  and 
a  measure  of  itself,  and  a  multiple  of  all  expressions  that  are 
free  from  this  letter;  and  every  such  expression  is  a  measure 
of  any  entire  function. 

QUESTIONS. 

1.  Write  an  entire  function  of  y  of  the  4th  degree,  whose 
coefficients  are  all  negative  integers. 

So,  an  entire  function  of  :(?,  and  one  of  x-\-y. 

2.  If  7^  be  a  fraction,  is  %:&  —  {ii  —  V)x^  +  (ii  —  2)a^  —  (w  —  3)a^ 
an  entire  function  ot  x? 

If  71  be  a  fraction  and  m  an  integer,  is 

nx^-\-2nx"'-^+inx'^-^+  ...     an  entire  function  of  a;  ? 
x""  +  anix'' - 1  +  hmH'' - ^  ?     «  +  Ix"^  +  cx^  +  dx"^  +  "  +  ca;'"»  ? 

3.  If  n  be  an  integer,  on  what  two  conditions  is 

aa:"  +  ^a:(»-i)/2  +  c2;("-3)/2  +  ^7^(«-5)/2    ^^  ^^^^^^  function  of  a;? 

r^n  _  ^^n  - 1  _f_  J^«  -  2  _^  ^^n  -  3  p       ^n  _^  ^^^(n  -  2)/2  _^  J^(n  -  4)/2  ^  ^^(n  -  6)/2  p 

4.  If  /2:=ia;2  +  3.r  +  6,  write  an  expression  for  /«,  f(-y}; 
and  find  the  values  of    /  l,/0,/-2,/5,/-5, /.167. 

6.  When  functions  of  a  single  letter  are  under  discussion 
and  an  expression  is  said  to  be  free  from  the  letter  of  arrange- 
ment, what  do  you  know  about  the  nature  of  that  expression  ? 
Can  you  tell  whether  it  is  an  integer  or  a  fraction  ?  whether 
it  is  positive  or  negative?  whether  it  involve  powers?  or  roots? 

6.  If  an  entire  function  of  a  single  letter  be  divided  by  any 
expression  free  from  the  letter  of  arrangement,  how  do  the 
exponents  of  the  letter  of  arrangement  in  the  several  terms  of 
the  quotient  compare  with  those  of  the  given  function  ? 

Why,  then,  is  every  such  expression  a  measure  of  every  such 
entire  function  ? 

What  integer  is,  for  a  like  reason, a  measure  of  all  integers? 


110  MEASURES   AND   MULTIPLES.  [IV,  Th. 

COMMON   MULTIPLES   AND   MEASURES. 

If  the  same  entire  function  be  a  multiple  of  two  or  more 
entire  functions,  it  is  a  cunn)io?i  multiple  of  them;  if  a  meas- 
ure of  them  all,  it  is  their  common  meastire. 
E.g.,     '7a^(y^^  —  l)f    iiy^  —  ^Y    are  common  multiples  of 

y  +  1,    y-1,    y'  +  l,    y'-l,    y'-y  +  h    ?/'  +  ?/  +  1 
and       x  —  a  is  a  common  measure  of  x^  —  a^,  a^  —  a^,  x^  —  aK 

That  common  multiple  of  two  or  more  entire  functions  whic7i 
is  of  the  lowest  degree  is  their  loivest  common  multiple;  and 
that  common  measure  which  is  of  highest  degree  is  their  high' 
est  co7nmon  measure. 
E.g.,    ^^  —  1     is  the  lowest  common  multiple  of 

y  +  \^y-l,   y^-\,y^-yJ,\,   /  +  i/  +  l,   y^ +  \,  y^ -1, 

and       x^  —  a^     is   the  highest   common   measure   of    x^  —  a*, 
x^  —  rt^,     a;*  —  a^,    x^  —  a^;    hu  t  not  of    x^  —  a^,    x^  —  a\ 

Ax.  2.    The  sum,  the  difference,  and  the ptroduct  of  tico  en- 
tire functions  of  a  letter  are  entire  functions  of  that  letter, 

Theor.  6.  A  common  measure  of  two  or  more  entire  func- 
tions of  a  letter  is  a  measure  of  their  sum. 
For,  let  r/a;"  +  Z>2;"  -  H  •  •  •  +  Jcx  +  /,  a'x"^  +  h'x''' " ^  +  •  •  •  +  ^-'.t  +  V, 
be  two  entire  functions  of  x,  and  m  a  common  measure 
of  tliem; 
then  •.•  ax""  +  bx''-^-^  --  -  -\-hx-\-l  =  M'P, 

a'x"'-\-b'x'^-'+  •  . .  +^-'2;  +  r  =  M.Q, 
wherein  p,  q,  are  entire  functions  of  x,  U^JV' 

.-.  «a:"  +  ^a;"-^+  ..  .  -\-kx-\-I-ha'.'''  +  yx'''-'^+  •  •  •  +k'x  +  V 

=:M.P  +  M-Q  =  M-(P  +  Q); 

and  *.•  the  sum     p  +  q     is  an  entire  function  of  x,  [ax.  2. 

.*.  M  is  a  measure  of 

ax''  +  bx''-^+  '  .  .  +kx-\-l  +  a'x"'-\-yx'^-'^-{-  •  •  •  +Kr  +  /', 
and  so  for  three  or  more  functions.  Q.e.d.         [df.  msr. 

If  the  entire  functions     ax''-\-I)x^-'^+  •  •  •  +^•.r  +  /, 

a'a;"*  +  b'x'^-'^-\ +  k'x  +  /',•••     be  represented    by  A,  B, •  •  •, 

the  proof  of  theor.  1  applies  word  for  word  to  theor.  6. 


6,  §2]  ENTIRE   FUNCTIONS   OF   ONE   LETTER.  Ill 

Cor.  a  common  measure  of  two  entire  functions  of  a  tetter 
is  a  measure  of  the  sum  and  the  difference  of  their  7nultij)tes, 

QUESTIONS. 

1.  A  common  multiple  of  two  or  more  entire  functions  of 
one  letter  is  of  at  least  what  degree  ? 

A  common  measure  cannot  exceed  what  degree  ? 

2.  In  what  three  ways  can  two  entire  functions  of  a  letter 
be  combined  with  absolute  certainty  that  all  the  exponents 
of  the  result  will  be  positive  integers  ? 

3.  What  is  the  lowest  common  multiple  of  a  +  b/c,  a  —  i/c, 
as  entire  functions  of  a?  as  entire  functions  of  ^  ? 

4.  Following  the  line  of  argument  of  theor.  6,  show  that 
a^  —  rr     is  a  measure  of    (a^  - x^)  +  {a^  - ;/;^)  +  {a^  -  x^). 

What  is  p  ?  Q?  R?     What  is  the  letter  of  arrangement? 

5.  Represent  a^-x^  l>y  A,  ft^-a;^  by  B,  ft*-n'*  by  c, 
and  write  out  the  proof  of  theor.  6. 

6.  In  the  proof  of  tlieor.  G,  if  3i  be  a  function  of  the  nW\ 
degree,  what  is  p? 

if  M  be  of  (;i  — 2)nd  degree,  of  what  degree  is  p? 
if  n>m,  what  is  the  highest  degree  possible  to  M  ? 
if  Q  be  free  from  .r,  what  is  the  degree  of  M  ? 
if  p,  Q  be  both  free  from  x,  what  relation  have  m,  n  ? 
What  is  the  degree  of  p  +  q? 

7.  A  common  measure  of  two  or  more  entire  functions  of 
a  letter  is  a  measure  of  their  sum,  their  difference,  and  their 
product:  is  it  also  a  measure  of  their  quotient? 

Illustrate  these  principles  by  aid  of  the  functions  in  ex.  3. 

8.  A  multiple  of  a  multiple  of  a  number  is  a  multiple  of 
the  number;  a  measure  of  a  measure  of  a  number  is  a  measure 
of  the  number;  and  a  common  measure  of  two  or  more  num- 
bers is  a  common  measure  of  any  multiples  of  them. 

9.  A  common  multiple  of  two  or  more  numbers  is  not 
necessarily  a  multiple  of  their  sum;  but  the  sum  of  two  such 
common  multiples  is  a  common  multiple  of  the  numbers. 


112  '    MEASURES  AND  MULTIPLES.  [IV,Th 

Euclid's  process  for  finding  the  highest  common 

MEASURE. 

Theor.  7.  If  the  higher  of  two  entire  functions  of  a  Utter 
be  divided  by  the  loiver,  the  common  measures  of  the  divisor 
and  the  remainder  are  the  common  measures  of  the  functions. 

If  the  loiver  function  he  divided  ly  the  remainder,  this 
divisor  by  the  second  remainder,  and  so  on,  then  some  re- 
mainder is  zero. 

The  last  divisor  is  the  highest  comtnon  measure  of  the  two 
e)i  tire  functions. 

Let  A,  B  be  two  entire  functions  of  a  letter,  Qi  the  quotient  of 

A  by  B,  and  Ri,  Rg, •  •  •  R^  the  successive  remainders; 
then  •.•  Ri  =  A  -  B  •  Qi ,  [df.  rem. 

.*.  the  common  measures  of  A,  B  are  measures  of  Ri ; 

[th.  G  cr. 
i.e.,      the  common  measures  of  A,  B  are  measures  of  B,  Rj. 

So,     *.•  A  =  B-Qi  +  Ri, 

.-.  the  common  measures  of  B,  Rj  are  measures  of  a; 
i.e.,      the  common  measures  of  B,  Rj  are  measures  of  A,  b; 
.•.  the  common  measures  of  A,  B  are  the  common  meas- 
ures of  B,  Rj,  and  there  are  no  others,      q.e.d. 

And  •/  Ri,  R2,  •  •  •  are  all  entire  functions  of  one  letter,  and 
of  lower  and  lower  degree, 
.*.  some  one  of  them  is  of  zero  degree,  as  to  that  letter, 
and,  if  not  itself  zero,  the  next  remainder,  Rn,  is  zero. 

Q.E.D. 

And  •/  the  common  measures  of  b,  Ri  are  the  common  meas- 
ures of  Rj ,  Rg ,  [above. 
.•.  the  common  measures  of  A,  B  are  the  common  meas- 
'               ures  of  Ri,  R2,  and  so  on; 

•*.  the  common  measures  of  A,  B  are  the  common  meas- 
ures of  R„-2,  R„-i ; 
and  *.•  R„_i  is  the  highest  common  measure  of  R„_o,  R„_i, 
.*.  R„_iis  the  highest  common  measure  of  A,  B.       q.e.d. 


T,§2]  ENTIRE   FUNCTIONS   OF  ONE  LETTER.  113 

If  R„_i  be  free  from  the  letter  of  arrangement,  then  A,  b 
are  usually  said  to  have  no  common  measure;  for  expressions 
free  from  the  letter  of  arrangement,  being  measures  of  all 
entire  functions  of  that  letter,  are  not  cliaracteristic  common 
measures  of  A,  B. 

Cor.  Every  remainder ,  Ei,R2,'''Ka  is  the  difference  of 
ttuo  7}iuUij)Jes  of  the  entire  functions,  A,  B. 

QUESTIONS. 

1.  In  the  jDroof  of  theor.  7,  if  the  higher  function  be  of  the 
mi\\  degree  and  the  lower  of  the  nth,  of  what  degree  is  the  first 
quotient?  the  product  of  the  divisor  by  this  quotient? 

What  is  the  highest  degree  possible  to  the  first  remainder? 
to  the  second  quotient?  to  the  second  remainder? 
Why  can  no  remainder  have  a  negative  exponent  ? 

2.  The  common  measures  of  b,  Ri  are  common  measures, 
and  the  only  common  measures,  of  Ri,  e.^  :  state  the  proof. 

3.  If  two  entire  functions  of  a  letter  have  an  integer  as 
a  common  measure,  is  this  measure  a  common  factor  of  all 
the  coefficients? 

4.  What  common  measure  free  from  x  have  the  functions 

Sax^  -  IQctJ^  +  24rf  -  12ax  -  4r^ 
24rtr6-*  -  Uaci?  +  Q^ax^  -  24r^.r  -  20^  ? 
Divide  out  this  common  factor  and  find  by  Euclid's  process 
the  highest  common  measure  of  the  quotients. 

5.  If  all  common  monomial  factors  of  the  two  functions 
have  been  divided  out  before  applying  Euclid's  method,  what 
does  a  remainder  free  from  the  letter  of  arrangement  show 
about  the  original  functions? 

6.  Show  that     A  =  B-Qi  +  Ri,     B  =  Ri.Qo  +  R.2,     Ri^Rg-Qa  +  Ro. 

Counting  Ej,  R2,  Rg  as  unknown  elements,  solve  these 
equations  for  R3,  and  show  that  its  value  is  the  difference  of 
two  multiples  of  A,  b. 

Can  all  other  remainders  be  expressed  as  such  differences  ? 

7.  In  ex.4  express  Ri,  r^,  R3  as  the  differences  of  two 
multiples  of  the  given  functions. 


114  MEASURES  AND  MULTIPLES.  [IV.Ths. 

PRIME   FUNCTION'S. 

An  entire  function  of  a  letter  that  has  no  measures  but 
itself,  and  expressions  that  are  free  from  that  letter,  is  a  prime 
function;  and  two  entire  functions  of  a  letter  that  have  no 
common  measure  except  expressions  that  are  free  from  that 
letter  are  prime  to  each  other. 

An  entire  function  of  a  letter  that  has  another  entire  func- 
tion of  that  letter  as  a  measure  is  a  composite  function,  and 
the  measuring  functions  are  its  factors. 
E.g.,    o?-\-x^\y    ^x^  —  l    are  prime  functions  of  a:, 
and      a?  +  x^  —  2x    is  prime  to  each  of  them,  though  itself  com- 
posite, having  the  factors    X,     x  +  2,    x  —  1, 

Theor.  8.  If  ttoo  entire  functions  of  a  letter  le  prime  to 
each  other,  then  two  multiples  of  them  can  he  found  such  that 
their  difference  is  free  from  that  letter,  and  conversely. 
For,  let  A,  B  be  two  functions  of  a  letter  that  are  prime  to  each 

other,  and  R„_i  be  their  highest  common  measure  ; 
then  *.*  R^_i  is  the  difference  of  two  multiples  of  A,  B,  say 
mk,  nB,  and  is  free  from  the  letter  of  arrangement, 
/.  ?w-A  — W'B  is  free  from  the  letter  of  arrangement. 

Conversely :  let  m-A  — ?^•B  be  free  from  the  letter  of  arrange- 
ment, and  not  0; 
then  •.*  every  common  measure  of  A,  B  is  a  measure  of  an  ex- 
pression that  is  free  from  the  letter  of  arrangement, 
.*.  A,  B  are  prime  to  each  other.  Q.  e.  d. 

Theor.  9.  If  an  entire  function  of  a  letter  le  prime  to  two 
or  more  such  functions,  it  is  prime  to  their  product. 
Let  A,  B,  c,  •  •  •  be  any  entire  functions  of  a  letter,  and  p  an 

entire  function  that  is  prime  to  each  of  them; 
then  is  p  prime  to  the  product  a-b-c-  •  • 
.For,  take     m,  n,    p,  q,     r,  s,  -  -  *     entire  functions  of  the 
letter  such  that    m-v  —  ^i-Afp-p  —  q-v,    r-p  — s-c,--- 
are  free  from  the  letter  of  arrangement;  [th.  8. 


8, 9,  §2]         ENTIRE    FUNCTIONS  OF  ONE  LETTER.  115 

then  •.•  the  product  (m-'P  —  7i'A)-{p-T  —  q-B)-(r-F  —  S'G)-'' 

is  free  from  the  letter  of  arrangement, 
i.e.,       h-F  +  k'A'B-G''-     is  free  from  the  letter  of  arrange- 
ment; wherein  h-F  is  the  sum  of  all  the  terms  that 
contain  p,  and  k=  dzii-q'S-  > ', 
,*  p  is  prime  to  the  product  a-b-c*  •  •  q.e.d.  [th. 8  conv. 

QUESTIONS. 

1.  Are  a^  +  S,  a^  —  S  prime  functions  of  a?  are  they  prime 
to  each  other  ?  what  is  the  letter  of  arrangement  ? 

2.  What  two  common  measures  belong  to  all  integers  ?  how 
then  can  integers  be  called  prime  to  each  other  ? 

3.  All  entire  functions  have  many  common  measures  that 
are  ignored  in  the  definition  of  functions  prime  to  each  other: 
what  are  these  measures?  why  are  they  ignored ? 

4.  In  Euclid's  process  for  finding  the  highest  common 
measure  a  remainder  of  unity  corresponds,  in  integers,  to  an 
expression  free  from  the  lettei  of  arrangement,  in  entire 
functions. 

5.  In  the  proof  of  the  converse  of  theor.  8,  if  the  expression 
m-A.  —  n-B  be  free  from  the  letter  of  arrangement,  every  com- 
mon measure  of  A,  B  is  free  from  tliat  letter:  state  the  proof. 

6.  In  the  proof  of  theor.  9,  what  relation  has  h-Fto  p? 
^  •  A  •  B  •  c  •  •  •  to  the  product  a  -  b  •  c  •  •  •  ? 

How  does  the  last  statement  depend  on  the  one  before  it? 

7.  If  there  be  two  entire  functions  of  a  letter  such  that  one 
of  them  is  a  measure  of  the  other,  an  entire  function  of  this 
letter  that  is  prime  to  both  of  them  is  prime  to  their  quotient, 

8.  If  an  entire  function  of  a  letter  be  prime  to  another  such 
function,  it  is  prime  to  any  integer  power  of  that  function. 

9.  From  the  fact  that  a  common  measure  of  two  numbers 
measures  their  sum,  prove  that  2  is  a  factor  of  any  number 
ending  in  0,  2,  4,  6,  8;  that  4  is  a  factor,  if  a  factor  of  the  last 
two  figures,  and  8,  if  of  the  hist  three  figures;  and  that  the  re- 
mainder got  by  dividing  the  last  figure  of  a  number  by  5  is 
the  same  as  that  for  the  entire  number. 


116  MEASURES  AND  MULTIPLES.  [IV,Ths. 

Cor.  1.  If  two  entire  functions  of  a  letter  he  prime  to  each 
other,  so  are  their  2^ositive  intcfjer  2)0ivers, 

Cor.  2.  If  an  entire  function  of  a  letter  measure  the  jyroduct 
of  two  such  functions  and  be  prime  to  one  of  them,  it  measures 
the  other. 
Let  A,  B  be  two  entire  functions  of  a  letter,  and  p  an  entire 

function  that  measures  their  product,  and  is  prime  to  a, 
and  let  mA,  nv  be  multiples  of  A,  P  such  that    7nx  —  nv—ly 

wherein  I  is  free  from  the  letter  of  arrangement;  [th.  8. 
then     m/l'K-nll'Y-l   and   w^//•A.B-?^/^P•B  =  B. 
and  *.-  P  measures  both  a-b  and  p-b,  [%Pv  df.  msr. 

/.  p  measures  B.  Q.E.D.         [th.  6  cr. 

Cor.  ?>.  If  a  prime  function  of  a  letter  7neasure  the  product 
of  two  or  more  entire  functions  of  that  letter,  it  measures  at 
least  one  of  thern. 

Cor.  4.  //  a  prime  function  of  a  letter  measure  a  positive 
integer  power  of  an  entire  function  of  that  letter,  it  measures 
the  function;  and  if  tlie  function,  then  tlie  poicer. 

Cor.  5.  If  an  entire  function  of  a  letter  be  measured  by  ttvo 
such  functions  that  are  prime  to  each  other,  it  is  measured  by 
their  product. 

Theor.  10.  A7i  entire  f^tnction  of  a  letter  can  be  resolved 
into  factors  that  are  prime  functions  in  but  one  way. 
Let  N,  an  entire   function  of   one   letter,   be   the   product 
a^-b^'-c"  •  •  •    wherein  a,  b,  c,  •  •  •    are  prime  functions 
of  that  letter  and     a,b,  c,  -  - '     are  positive  integers; 
then  N"  has  no  other  prime  factor. 

For  •.*  any  other  prime  function,  F,  is  prime  to  each  of  the 

prime  factors  A,  B,  c,  •  •  •  ['lyp- 

and  to  their  powers    A",  B^  C,  •  •  •  [th.  9  cor.  1. 

.*.  F  is  prime  to  the  product  N".  [th.  9. 

So,  A  can  be  a  factor  of  N  not  more,  and  not  fewer,  than  a 
times;  and  so  with  b,  and  witli  the  other  prime  factors. 


9,10,12]  ENTIRE  FUNCTIONS  OF  ONE   LETTER.  117 

Cor.  a  common  measure  of  huo  or  more  entire  functions  of 
a  letter  can  contain  no  factor  that  is  not  in  all  of  them, 

QUESTIONS. 

1.  In  the  proof  of  theor.  9  cor.  2,  if  mfl,  n/l  be  fractions, 
"how  can  b  be  a  multiple  of  p  ? 

2.  Show  that  cor.  4  is  only  an  application  of  cor.  3  to  factors 
having  a  special  relation  to  each  other. 

3.  In  cor.  5,  let  A  be  an  entire  function  of  a  letter,  and  let 
p,  Q  be  entire  functions  prime  to  each  other,  but  measures 
of  a:    if  A=:?i-Q,  what  relation  has  n  to  p? 

If  n  —  m-v,  what  is  A  in  terms  of  m,  p,  Q? 

4.  A  measure  of  an  entire  function  of  a  letter  contains  no 
factor  that  is  not  a  factor  of  the  function;  and  a  common 
measure  of  two  or  more  such  functions  contains  no  factor 
that  is  not  in  all  the  functions. 

What  are  the  factors  of  the  highest  common  measure  ? 

5.  If  F  be  a  factor  of  an  entire  function  of  a  letter,  f"  is  a 
factor  of  the  nih.  power  of  the  function. 

6.  An  entire  function  of  a  letter,  if  measured  by  any  num- 
ber of  such  functions  that  are  prime  to  each  other,  is  measured 
by  their  product;  and  the  product  of  all  the  prime  factors 
common  to  two  or  more  such  functions  is  a  common  measure 
of  the  functions. 

7.  If  a,  h,  c,  ' ' '  I  be  any  entire  numbers  and  m  another, 
if  a\  V ,  c',  •  •  •  V  be  the  remainders  when  «,  ^,  c,- '  -l  are 
divided  by  m,  and  if  the  sum  «'  + J'  +  c'+  •  •  •  4-^  be  meas- 
ured by  m,  so  is  the  sum     a-\-h  +  c+  >  •  -  +1, 

If  3  be  a  factor  of  the  sum  of  the  digits  of  an  integer,  it  is 
a  factor  of  the  integer;  and  so  with  9. 

8.  In  the  last  part  of  the  proof  of  theor.  10,  suppose  the 
factor  A  to  occur  fewer  than  a  times :  what  are  the  only  factors 
by  which  it  can  be  replaced  ?  Can  a  power  of  one  of  the 
other  factors  take  the  place  of  a  power  of  A?  If  a  were  used 
more  than  a  times,  what  change  must  occur  among  tne  other 
factors  ?     Is  that  possible  ? 


118  MEASURES   AND   MULTIPLES.  [IV,  Pbs 

§  3.   ENTIRE  FACTORS. 

Integers  and  entire  functions  of  a  letter,  or  letters,  may  be 

classed  together  as  entire  numbers. 

The  measures  of  an  entire  number  are  its  entire  factors. 
The  2)f'inie  factors  of  a  composite  entire  number  are  tlie 

prime  entire  numbers  whose  product  is  the  given  number; 

and  to  factor  a  number  is  to  find  all  its  prime  factors. 

E.g.,    GOOa'x^  -  600a^x'  =  2^  •  3  •  5^  •  a^  •  rz;^  •  («  +  :c)  •  («  -  x), 

PrOB.  1.    To    FACTOR   AN"   INTEGER. 

Divide  the  numher,  and  the  successive  quotients  in  order,  hy 
the  pri?nes  2,  3,  5,*  •  •,  usi^ig  each  divisor  as  many  times 
as  it  measures  the  successive  dividends. 
The  successful  divisors,  and  the  last  undivided  dividend, 

are  the  prime  factors  sought;  and  no  divisor  larger  than  the 

square  root  of  the  dividend  need  be  tried. 

For  the  dividend  is  the  product  of  the  divisor  and  quotient, 

and  if  the  divisor  be  larger  than  the  square  root  of  the  divi- 
dend, then  the  quotient  is  smaller; 

i.e.,  if  there  be  a  factor  larger  than  the  square  root  of  the 
dividend,  there  is  also  a  smaller  factor; 

and  all  the  possible  smaller  factors  have  been  already  found. 

If  a  number,  not  a  perfect  square,  have  no  factor  smaller 
than  its  square  root,  it  is  prime. 

E.g.,  of     11908710,     2  is  a  successful  divisor  once,  3  twice,  5 
once,  11  once,  23  once,  and  the  square  root  of  the  quo- 
tient, 523,  is  smaller  than  23; 
/.  the  prime  factors  are     2,  3,  3,  5,  11,  23,  523. 

PrOB.  2.    To  FACTOR  A  POLYNOMIAL  OF  KNOWN  TYPE-FORM. 

Express  the  number  in  one  of  the  typeforms,  and  write  its 

factors  directly  in  the  form  of  the  factors  of  the  type. 
E.g.,     r*  +  2aa;  -f  a^  -  2bmhi^  =  {x  +  aY  -  {bm  nf, 

=  {x-\-a  +  bmn)'{x  +  a  —  bmn).  [II.  pr.  3  nt.  4. 


1,2.  §3]  ENTIRE  FACTORS.  119 

QUESTIONS. 

1.  Make  a  table  of  the  prime  numbers  from  0  to  100. 
Factor,  or  prove  to  be  prime : 

2.  30;   37;  72;   120;   323;   367;   1331;   1683;   8279;   15625. 

3.  «2  +  2«^  +  5l  4.  7?i^ +  2771  +  1.  5.  a^-2xy  +  y\ 
6.  x'  +  DX  +  6.             7.  4:a^  + 2a -20.         8.  m^-7i\ 

9.  4:m^-9xK  10.  a^-y\  11.  a^m.^n^ 

12.  xY-16yhK        13.  x^'-ij''.  14.  e^-e-^. 

15.  ^dz2  +  e-K        16.  ««-2565-8.  17.  a*a;2  +  a'^-al 

18.  a'-c\  19.  276-^  +  1.  20.  a;^- 13a; +  42. 

21.  2.^2  +  6.^-8.         22.  a;*-m-/;2  +  9.        23.  if-y-30. 
24.  ft2_4^_32.         25.  a^-8^,  26.  4/-2?/  +  l. 

27.  4«2_4^^j_}.j2^       28.   {x±yY-z\         29.  rt3_^2^_6^^2^ 
30.  12a'  +  a^x^-xK    31.  6-2-10c^/  +  25fZl  32.  a^^n  +  Orr Z*  +  ^l 
33.  x-^  +  6x-'^  +  9.      34.  a;-2-5.c-H6.      35.  a-^"'-2a-'^b  +  b\ 
36.  9.7.-2  +  3fl!.-c  +  6.c  +  2«.  37.  a7+lV  a  +  3  •  ^+1  +  3^. 

38.  x^  +  2xy+?f  +  5x  +  Dy  +  6.        39.  («-«-i)8_(^,_^,-i)8. 
40.  {a  +  xY-{a-C'')\  41.  «2.^-2_3^^3.^_^2ri*. 

42.  2-3+ ?/H  3.7; ?/ (;.'  +  //).  43.  a'-b'-3ai{a-i). 

44.  5(r/;2-?/2)  +  30(;+//f.  45.  3(a^-y^)-5{x-yY. 

46.  2(aH«'^  +  rt^')-(a^-<^').        47.  «*  -  J*  +  (r^2  _  ^2^2^ 
48.  2x^y  +  53fy'  +  2xy\  49.    6/ -  3.ry  -  9.?:^. 

50.  «2_§2_^_2Jc.  51.   ac  +  ad  +  hd  +  bc, 

52.  a;2P4.(^;  +  j)2;P  +  orJ.  53.  a^i'-(«  +  ^,)a;P  +  «J. 

54.  a;2^  +  (a-^')2;^-«J.  55.  afP-{a-b)xP-ab. 

56.  2;^  +  2/^  +  2^  —  2.T?/  ±  2xz  T  2^2.  [observe  the  signs. 

57.  2a'b'  +  2a'c'  +  2b'c'-a'-b'-c',  [Aa^^ - (a^  +  b' - (f)\ 

58.  a'^  +  ^b''  +  9c^+  •  •  •  +4rt^  +  6fl^c+  •  •  •  +12bc+  ...  +  ... 

59.  m^  —  n^  —  ???  (??z^  —  ?i^)  +  w(??i  —  ??)^  +  (??/^  +  ?r)  (??i  —  w). 

60.  a2-«5-2(r/J-?>2)  +  3(rt2_^2)_4(^^_j)2^ 

61.  (a;  +  ^)2  +  2(;.^+.TZ/)-3(:/;2-/)  +  4(a:?/+/j, 


120  MEASURES   AND   MULTIPLES.  [IV,  Pr, 

PrOB.  3.    To  FACTOR  AK  ENTIRE  FUNCTION  OF  ONE  LETTER. 

Find  the  common  factors  of  the  coefficieiits  and  divide  them 

out;  .  [th.  6  cr. 

hy  trial,  or  hy  comparison  with  hioivn  typeforms,fi?id  a  factor 

of  degree  not  higher  than  half  the  degree  of  the  function. 
If  all  the  coefficients  be  integers,  try  no  factor  unless  its  first 

and  last  coefficients  measiire  the  first  and  last  coefficients 

of  the  function; 
try  710  factor  unless  its  value  measures  that  of  the  function 

when  the  letters  have  convenient  values  given  to  them  ; 
if  all  the  coefficients  in  the  function  he  positive,  try  no  factor 

whose  first  and  last  coefficients  are  not  both  positive, 

E.  g.,  of    A:Qax^  + 1  '^Oaxy  +  Ihay^    the  factors  are 
5,  a,  8a;H26x?/  +  15/; 

and  *.•  1,  2,  4,  8  are  the  measures  of  8,  and  1,  3,  5,  15,  of  15, 
and  all  the  coefficients  are  positive, 
/.  the  possible  factors  of     ^x^  +  2Qxy  +  15y^    are 
x  +  y,  2x-hy,  4:X  +  y,  8x  +  y, 

x-\-3y,  2x  +  3y,  4:X  +  3y,  8x  +  ^y, 

x  +  by,  2x  +  5y,  Ax  +  5y,  Sx  +  5>/, 

x  +  lby,  2x  +  lby,  4:X  +  15y,  8x  +  l5y. 

In    Sx^  +  2Qxy  +  16y^,    and  in  these  sixteen  possible  measures 
put    x  =  l,    y  =  l; 

then     Sx^  +  2Qxy  +  16y^  =  4:d,     whose   measures   are  1,  7,  49, 

and  only    4:X  +  ^,  =7,     and     2x-^6y,=7,     pass  this  test; 

and,  by  multiplication,     4:X  +  3y,     2x-{-5y    are  found  to  be 
the  factors  sought. 

So,  of     7.^^ -  30x^  +  62x  -  45,    the  possible  linear  factors  are 

x±l,        x±^,        x±5,        a- ±9,        a;  ±15,        x±45, 
7a;  ±1,       7a;  ±3,      7a;  ±5,      7a;  ±9,      7a;  ±15,      7a;  ±45. 

In     7a;^- 30a;'' 4- 62a; -45,     and  in  these  factors,  put     a;=l; 

then  the  function  is  -  6,  and  the  only  possible  factors  of  it  are 
x-\-l,     x-S,    a;  +  5,     7a;-l,     7a;-5,     7a;-9. 


3,  §3)  ENTIRE  FACTORS.  121 

So,  put  x  =  '2;  then  the  function  is  15,  and  out  of  the  six 
possible  factors  above  the  only  ones  still  possible  are 
x  +  1,     x-3,     7x-9; 

and  *.•  of  these  three  factors  7a:  —  9  succeeds,  and  the  others  fail, 
.',  7a;  — 9,    a^  —  3x-\-b     are  the  factors  sought. 

QUESTIOXS. 

1.  In  factoring  an  entire  function  of  one  letter,  why  need  no 
factor  of  degree  higher  than  half  that  of  the  function  be  tried  ? 
what  direction  in  the  rule  for  factoring  integers  is  like  this  ? 

2.  Of  what  two  terms  is  the  first  term  of  the  dividend  the 
product  ?  the  last  term  of  the  dividend  ? 

3.  Is  ci^  +  x  +  1  a  factor  of  a^  —  1  for  certain  values 
only  of  X  or  for  all  values? 

In  these  functions  replace  x  by  1,  2,  3,  •  •  •  in  turn,  and 
show  that  the  first  result  measures  the  second  in  every  case. 

4.  Let  x^-\-H^  +  cx-{-d  be  an  entire  function  of  x,  and 
b,  c,  d  be  all  positive;  divide  by  x  —  a\  show  that  a  positive 
remainder  is  left  at  every  step  of  the  process;  and  that,  there- 
fore, a  function  whose  coefiicients  are  all  positive  has  no 
measure  of  the  form     x  -  a. 

Factor,  or  prove  to  be  prime: 

5.  6.7;* +  2.^^-12/.  6.  QbH^-nx^-'^:!^, 

7.  6;/r+(2rt  +  l):6'~(«  +  2).  8.  iDx'^  +  Sa^y-^Uy^-lbyK 

9.  ahx^  +  a^x  +  V^x  +  ah.  10.  a^T^-y^y^  +  c^z^. 

11.  a^:f^  +  2a^hx'  +  2aWx  +  h\  12.  a'^x^-^a'b^x''  +  hK 

13.  ^a^  +  Qabz-4.acz-Sbcz\  14.  4:X^-{%^  +  lU^)x'  +  ^M^. 

15.  x^-\-{a  -b  +  c  —  d)j?  +{—ab  +  ac  —  ad  —  bc  +  bd  —  cd)x^ 

+  ( —  abc  +  abd  —  acd  +  bcd)x  +  abed. 

16.  aby^  +  {ci"  +  a'b^  +  b'')y^  +  {nb  +  a^b  +  ab^)y  +  ct'bK 

17.  45a:3  +  83:/:2^-100.r?/-49/y3.  18.  7.7:3-25.^2  +  11.1' +  3. 
19.  5.v;2  +  17.r  +  3.  20.  '^x^-IOx'^  +  ^x  +  d.  21.  a''±ab  +  b\ 
22.  a^±a^b  +  ab^±b\                   23.  18«3-24«2_19«  +  18. 

34.  S:(?-2Qx'  +  2^x-V2,  25.  12a:*-16a:3-lla:«-8a:-42. 


122  MEASURES  AND  MULTIPLES.  [IV,  Th 

LINEAR   FACTORS. 

First  degree  factors  are  linear  factors. 
E.g.,  x-a,y^z,z  —  Z-{-c, 

Theor.  11.  If  fx  le  an  entire  function  of  x,  then    x—a 
meas  u  res    fx  —fa. 
For,  let    fx  =  ^-\-Bx  +  cx^-\ 1- La;",     wherein    A,  b,  c  •  •  •  L 

are  free  from  x,  but  may  contain  other  letters  and 

numerals; 

then    /^^  =  A  +  Brt  +  C«^  H +  L«", 

and    fx  -fa  =  B{x-a)  +  c{x^-a^)-\ +  L(a;"  -  «"), 

which  is  measured  by    x  —  a,  Q.E.D. 

Co^,  1.  If  fx  le  divided  hy    x  —  a,    the  remainder  is  fa. 
For    fx  =  B(x-a)+c{x^-a^)+'  •  •  +L(a;"-c^")+>, 
and  each  term  of  the  function  in  this  form  is  divisible  by 
x  —  a,    except /rt,  which  is  free  from  x,  and  is  the  re- 
mainder. Q.E.D. 

Cor.  2.  Iffa  —  ^,  then  x  —  a  measures  fx,  and  conversely. 

From  this  corollary  comes  a  new  rule  for  finding  linear 

factors  of  a  function  of  one  letter: 

In  the  function,  replace  the  letter  of  arrangement  hy  any  num- 
ber; if  the  function  is  thereby  made  zero  the  letter  of 
arrangement  less  this  numler  is  a  factor. 

E.g.,  to  factor    ic*  -  Sx^  +  ^x^  +  38a:  -  40 : 

put  1  for  a;;  then  /c=:  1-8 +  9 +  38 -40  =  0,  and  x-1  is 
a  factor,  with  the  quotient     a;^  —  7a;^  +  2a;  +  40 ; 

put  1  for  X  in  this  quotient;  then  /ia;=:  1  —  7  +  2  +  40  =  36, 
and    a;  — 1     is  not  again  a  factor; 

put  2  for  a;;  then  /ia'  =  8-28  +  4  +  40  =  24,  and  a;-2  is 
not  a  factor; 

put -2  for  a;;  then   /ia;= -8-28-4  +  40  =  0,    and    a-  +  2isa 

factor,  with  the  quotient     .r^-9a;  +  20, 
whose  factors  are    a;  —  4,  a;  —  5. 

.•.a;*~8.r'  +  9a;2  +  38a;-40  =  (a;-l).(a;  +  2).(.f-4).(:?;-5). 


il,§3]  ENTIRE   FACTORS,  123 

QUESTIONS. 

1.  If  in     x^  —  a"^     X  be  replaced  by  «,  what  does  the  value 
of  the  expression    ic"  — ^"  become  ?  what  does  this  prove  ? 

So,  if  X  be  replaced  by  ~a,  when  n  is  even? 

So,  if  X  be  replaced  by  "«,  when  n  is  odd  ? 

So,  if  in     »;"  +  «",     X  be  replaced  by  "of  when  n  is  odd? 

Is  x^-\-a^  divisible  by  either  x-a  ov  x  +  a  when  n  is  even  ? 

2.  State,  as  theorems,  all  the  conclusions  reached  in  ex.  1. 

3.  Divide    a?  —  Qx^  +  10x  —  S    by    x  —  2,    and  compare  the 
remainder  with/2. 

So,  divide  by    x  —  4:    and  compare  with /4. 

4.  If  fx  be  0  when  x  is  replaced  in  turn  by  a,  h,  c,  • » *  f 
then  fx  is  divisible  by  {x  —  a)-  (x  —  b)'(x  —  c)''» 

6,  x  —  a     is  a  factor  of 

(c^  +  2x  +  3)'{a^  +  a)-(a''  +  2a  +  3)'{x^  +  x). 
Find  the  linear  factors  of: 

6.  Gx^-'Tx'-Sx  +  ie.  7.  a*  +  4«3  +  4rtH4«  +  8. 

8.  a;*  +  4a;^-25a;2-16a;  +  84.  9.  a;3-8a;Hl92;-12. 

10.  a^-7«2  +  Urt-8.  11.  /-4?/3-7?/Hl0y. 

12.  a:3  +  8.x2  4-202;  +  16.  13.  ;2;^  -  3^^  ^  10;^  -  8. 

14.  c*- 136^36.  15.  /-lly2  +  18y-8. 

16.  x^-a^-39x^-^24rX-\-180.        17.  a.'3  +  5a;2-92;-45. 
18.  a^-8«Hl3«-6.  19.  x'-x^-llx^-{-9x  +  18. 

20.  a;^  -  'Sx'  +  6.r3  -  3a.-2  -  3:c  +  2.     21.  3a^  -  6abz  +  iacz  -  8bcz\ 

22.  x^-2x^-lox^  +  8x'^  +  6Sx  +  4:S, 

23.  3(«  +  Z>)^  +  8(«  +  ^)H«  +  Z>-2. 

24.  ic^  -  (rt  -  J  +  c)a;2  +  (ac  -ah-  hc)x  +  «J6?. 

25.  x'iif  +  ?/«^'  +  2«a;^ - a;^^^ -  ?/^2;9 - z'^x^    is  measured  by 

{^-y)'{y  —  z)-{z  —  x)     if  g,  r  be  any  positive  integers. 

By  the  light  of  ex.  25,  find  the  factors  of: 

2 G.  x^y  +  y'^z  +  zH  —  xy'^  —  yz^  —  zi\ 

27.  a^y+  yh  +  z^x  —  xy^  —  yz^  -  zx^. 

28.  a^y"^  -h  fz^  -f  2 V  -  xy  -  yh^  -  z^xK 


124  MEASURES  AND  MULTIPLES.  CIV.Ths. 

FRACTIONS   IN   THEIR   LOWEST   TERMS. 

Theor.  12.  If  the  terms  of  a  siminle  fraction  he  prime  to 
each  other,  the  fraction  can  he  reduced  to  no  equivalent  shiqjle 
fraction  in  lower  terms. 
For,  let  a/b  be  a  fraction  such  that  A  is  prime  to  B,  and   p/q 

an  equal  fraction; 
then  •••  a/b  =  p/Q,  [hyp. 

.-.  A-Q=B-p; 
and   •.•  B  measures  A-Q,  and  is  prime  to  A,  [hyp. 

.*.  B  measures  Q.  [ths.  4,  9. 

So,        A  measures  p. 
and  p/q  is  not  in  lower  terms  than  a/b.  Q.  e.  d. 

Cor.  \,  If  a  fraction  he  in  its  loiuest  terms,  so  is  every 
integer  poiuer  of  it. 

Cor.  2.  A  fraction  in  its  lowest  terms  can  he  resolved  into 
hut  one  set  of  factors  and  divisors.  A",  B^  •  •  •  G^,  h'' •  •  •, 
wherein  a,  b,  •  •  -g,  h-  •  •  are  different  prime  numhers,  and 
a,  h,- '  -g,  h,'  • '     are  integers,  some  of  them  negative, 

FACTORS  OF  THE  HIGHEST  COMMON  MEASURE  OF  TWO  NUMBERS. 

Theor.  13.  The  product  of  all  the  common  prime  factors 
of  tiuo  or  more  mimbers,  each  taken  2oith  the  least  exjwnent  it 
has  in  anij  of  the  numhers,  is  the  highest  common  measure 
of  the  numhers. 

Cor.  1.  Every  common  measure  of  two  or  more  numhers  is 
a  measure  of  their  highest  common  measure. 

Cor.  2.  If  each  of  two  or  more  numhers  he  multiplied  or 
divided  hy  the  same  number,  their  highest  common  measure  is 
multijjlied  or  divided  hy  this  number. 

Cor.  3.  Tlie  highest  common  measure  of  two  or  more  num- 
bers is  not  changed  by  niulfiplyiug  or  dividing  either  mimher 
by  a  number  j)rime  to  any  of  the  others. 


12.  13,  §3]  ENTIRE   FACTORS  125 

i^UESTIOXS. 

1.  Wheu  is  it  desirable  to  cliange  a  given  fraction  to  higher 
terms?  how  can  it  be  done? 

2.  If  a  fraction  be  in  its  lowest  terms,  what  is  true  of  its 
numerator  and  denominator  ?  of  any  integer  powers  of  them  ? 

3.  Is  the  fraction  3289/3325  in  its  lowest  terms? 

4.  If  the  fraction  a/b  be  not  in  its  lowest  terms  but  A,  B 
have  the  single  common  factor  f,  by  what  single  process  can 
(a/b)"  be  reduced  to  its  lowest  terms  ? 

5.  The  entire  factors  of  a  fraction  are  factors  of  which 
part  of  it  ?  the  divisors  are  factors  of  which  part? 

In  how  many  ways  can  either  of  these  parts  be  factored  ? 
What  kind  of  numbers  are  the  factors  of  a  fraction  ? 
What,  the  reciprocals  of  the  divisors  ? 

Why  are  some  of  the  integers  a,  Z>,  •  •  •  A  of  theor.  12  cor.  2 
negative?  why  not  all  of  them  ? 

6.  Prove  theor.  13,  by  sliowing  that  such  a  product  measures 
each  of  the  numbers,  and  that  no  higher  number  can  measure 
them  all. 

7.  How  many  of  the  common  factors  of  several  numbers 
are  found  in  their  highest  common  measure  ? 

8.  In  how  many  of  the  numbers  must  a  factor  be  found  in 
order  to  be  a  factor  of  their  highest  common  measure  ? 

9.  If  a  common  factor  be  rejected  from  two  or  more  num- 
bers and  the  highest  common  measure  of  the  quotients  be 
found,  what  has  been  done  to  the  highest  common  measure  of 
the  given  numbers? 

How  must  the  highest  common  measure  that  has  been 
found  be  changed  to  give  that  of  the  original  numbers? 

10.  If  two  fractions,  when  reduced  to  their  lowest  terms, 
have  different  denominators,  their  sum  can  not  be  an  entire 
number. 

11.  If  the  denominator  of  a  fraction  in  its  lowest  terms 
have  other  factors  tlian  2  and  5,  the  fraction  can  not  be  exactly 
expressed  as  a  decimal. 


126  MEASURES    AND   MULTIPLES.  [IV,Th& 

FACTORS"  OF   THE    LOWEST    COMMON   MULTIPLE    OF   TWO    OR 
MORE    NUMBERS. 

Theor.  14.  Tlte  product  of  all  the  different  prime  factors 
of  two  or  more  entire  numbers,  each  ivitli  the  greatest  exponent 
it  has  in  any  of  the  numbers,  is  their  lowest  common  multiple. 

[df.  1.  c.  msr.,  th.  4  cr.  5,  th.  9  cr.  5o 

Cor.  1.  Every  common  multiple  of  two  or  more  numbers  is 
a  multiple  of  their  lowest  common  multiple. 

Cor.  2.  If  each  of  two  or  more  numbers  be  mnltiplied  or 
divided  by  the  same  mimbcr,  their  lowest  common  inultiple  is 
multijMed  or  divided  by  this  number. 

Cor.  3.  The  product  of  two  numbers  is  the  j^ro duct  of  their 
highest  common  measure  and  loioest  common  multiple. 

COMMON    measures   AND    MULTIPLES    OF   THREE    NUMBERS. 

Theor.  15.  The  highest  common  measure  of  three  numbers 
is  the  highest  common  measure  of  the  highest  common  measure 
of  any  two  of  the  numbers  and  the  third  number;  and  so  for 
the  loiuest  common  multiple, 

COMMON    MEASURES   AND    MULTIPLES    OF   TWO    FRACTIONS. 

Theor.  16.    TJie  highest  common  measure  of  tiuo  fractions 
in  their  lowest  terms  is  the  quotient  of  the  highest  conunon 
measure  of  the  numerators  by  the  lowest  common  multiple  of 
the  denominators;  and  their  loioest  common  multiple  is  the 
cjuotient  of  the  lowest  common  multiple  of  ilie  numerators  by 
the  highest  common  measure  of  the  denominators. 
For,  let  a/b,  c/d  be  two  fractions  in  their  lowest  terms, 
;uid  let  f/m  be  a  measure  of  them  in  its  lowest  terms; 
then  •.*  a/b:  f/m,  c/d:  f/m,  i.e.,  am/bf,  cm/df  are  entire, 

.*.  F  is  a  common  measure  of  A,  c,  and  m  a  common  mul- 
tiple of  B,  d;  [th.  4  cr.  2,  th.  9  cr.  2. 
and        f/m  is  highest  when  F  is  the  highest  common  measure 
of  A,  c,  and  m  the  lowest  common  multiple  of  b,  d. 


14.  15, 1G,§3]  ENTIRE   FACTORS.  127 

So,  let  m/f  be  a  common  multiple  of  a/b,  c/d; 

then  *.*  m/f:  a/b,  m/f:  c/d,  i.e.,  bm/af,  dm/cf  are  entire, 

/.  F  is  a  common  measure  of  B,  d,  and  m  a  common  mul- 
tiple of  A,  c,  [th.  4  cr.  2,  th.  9  cr.  2. 

and       m/f  is  lowest  when  f  is  the  highest  common  measure 
of  B,  D,  and  M  the  lowest  common  multiple  of  A,  c. 

QUESTIONS. 

1.  In  the  proof  of  theor.  14,  show  that  each  of  the  numbers 
is  a  measure  of  such  a  product,  and  that,  if  any  factor  were 
omitted  from  this  product  or  taken  fewer  times,  some  one  of 
the  numbers  would  no  longer  be  a  measure  of  it. 

2.  In  the  low^est  common  multiple  of  A,  B,  what  factors  of 
A  are  found,  and  what  factors  of  B  are  among  them  ? 

What  other  factors  must  be  added  ? 

3.  Let  A,  B  be  any  two  entire  numbers,  H  their  highest 
common  measure,  L  their  lowest  common  multiple,  and  let 
A  =  aH,  b  =  ^h;  let  a  have  the  factor  c"*  and  b  tlie  factor  c", 
and  m  >  n :  how  many  times  is  the  factor  c  in  H  ?  in  «  ?  in  L  ? 

If  m<n,  how  many  times  is  C  in  H?  in  Z*?  in  L? 

4.  A  highest  common  measure  found  as  in  theor.  15  contains 
all  the  factors  common  to  all  the  numbers  and  no  others;  and 
a  lowest  common  multiple  contains  all  their  factors. 

5.  In  the  proof  of  theor.  16,  if  the  fractions  be  multiplied 
by  the  lowest  common  multiple  of  their  denominators,  what 
do  the  fractions  become  ?  How  is  the  highest  common  measure 
of  these  products  related  to  that  of  the  given  fractions? 

If  the  fractions  be  in  their  lowest  terms,  can  the  lowest 
common  multiple  of  the  denominators  contain  any  factor  in 
both  numerators?  Are  the  multipliers  of  the  numerators 
prime  to  each  other? 

6.  Let  a/b,  c/d  be  two  fractions  in  their  lowest  terms,  and 
let  B  =  ^-F,  d  =  <:Z-f:  what  is  f?  what  is  b-d-F? 

Multiplying  both  these  fj-actions  by  ^'^-F  gives  two  integers 
whose  lowest  common  multiple  is  bd  times  the  lowest  common 
multiple  of  a,  c,  and  bdF  times  that  of  the  fractions. 


128  MEASURES  AND   MULTIPLES.  [IV,  Pr. 

§4.   THE    HIGHEST   COMMON   MEASUKE. 

PrOB.    4.    To    FIND    THE    HIGHEST    COMMON    MEASURE    OF 
TWO   OR  MORE   ENTIRE   NUMBERS. 

(a)  The  prwie  factors  of  all  the  munbers  knotvn:  multiply 
together  all  the  different  2^'t'i'tfie  f (Victors,  each  ivith  the 
greatest  exponent  it  has  in  all  of  the  numbers,    [tli.  13. 
E.g.,  of    9a^b%    3ab^cd,    15ab^c^  —  12ab^,    the  common  factors 
are  3,  a,  b^;  and  the  highest  common  measure  is  3ab'^. 
So,  •.•  2x'y  +  6xy^ - 6x^ - 2?/^  =  2 {x - y) {x  +  y)(y- ^x) 
IQax^y  —  12ax^  —  4:axy^=4ax{x  —  y)(y  —  ^x) 
10y^-60xy^-\-70x^y-dO:^=:lO{x-yY(y-3x), 
.*,  the  h.  c.  msr.  of  these  expressions  is  2(x  —  y){y  —  3x), 
{b)  T7ie  prime  factors  not  knotv7i,  tiuo  entire  numbers:  divide 
the  higher  number  by  the   lotver,  the  divisor  by  the 
remainder  if  any f  that  divisor  by  the  second  remainder, 
and  so  on  till  nothing  remains;  [ths.  2,  7. 

at  pleasare,  suppress  from  any  divisor  any  entire  factor  that 
is  prime  to  the  dividend  corresponding,  and  introduce 
into  any  dividend  any  entire  factor  that  is  prime  to 
the  divisor;  [th.  13  cr.  3. 

at  pleasure,  suppress  from  any  divisor  and  the  corresponding 
dividend  any  common  measure  of  them,  but  reserve  this 
measure  as  a  factor  of  the  final  result,        [th.  13  cr.  2. 
The  product  of  tlie  last  divisor,  as  above,  by  the  reserved  fac- 
tors if  any,  is  the  highest  common  measure  sought,   [ths.  2,  7. 
E.g.,  to  find  tlie  h.  c.  msr.  of    x'^  +  x-U,    a;2-10a;  +  21: 
write   a;H^-12)2;«-102;  +  21(l  or     1     1  "12)1  "10  +21 

x^+     x-\2  1       1  -12 

-ll)_-ll^+33  -11)-11     33 

a;«-3^ a;-3(a;  +  4  1  "3  1  "3 

4x-\2  4  -12 

znd    a;— 3    is  the  measure  sought. 

So,  of  4<^«rc^  +  4«2;  —  48^^     4^<:/-^  — 40<^w;  +  84rr,     4r?    is  a  common 
factor,     ic  — 3     is  the  li.c.msr.  of  the  remaining  factors, 
and     4rt(a;  — 3)     is  the  highest  common  measure. 


4,  §4]  THE  HIGHEST   COMMON  MEASUPwE.  129 

QUESTIO>^S. 
Find  the  highest  commou  measure  of: 

1.  :j-1,     cc'-h     ^'^-l'  2.  l-a-^     (l+r^f,     l+ar^. 

3.  Z{a^-(rx),     4(x''  +  ax),     D(x'-a'),     G{x'-a'). 

4.  a^-\-2x- 3,     x^ -73^  +  Gx.     5.  3.^•2 - 24:X - 9,     2.6-3 _iq^_  g, 

6.  l-2.r,     l-4.x\     1-^x?.     1-16./;*,     l-32:d 

7.  o^^x^  +  x^  +  x^'-^x  +  l,     x^'-x  +  l, 

8.  4  +  52;  +  r6^     S-2x-x\     U  +  lx  +  x"^,     20  +  x-q?. 

9.  529(^-2  + a; -6),     782 (2.1-2  + 7a; +  3),     935(2^2_3^_2), 
10.  .^■3  +  3.c2  +  4a;  +  12,     a.'3-5:c2^4.^,_2o. 

Reduce  these  fractious  to  their  lowest  terms: 

3?-Gx  +  b  l  +  3.7;-42:^-12.y* 

Ix'-l^x  +  b'  Sj^-^x'-^x  +  V 

l  +  .r^  +  25.r*  3^3-18^^^  +  33^-18 

l  +  3a;-15a;2-25a;*  *        i2f?-84r^  +  72     * 

15.  Given     3^^- 10^  +  3,     W  +  7b-20,     m^^-n^,     all   prima 

to  each  other:  find  the  highest  common  measure  of 
(3ft2 _  10«  +  3)3 .  (6Z/2  +  7^  -  20)2 .  (;7^H  ?i3^*, 
(3rt2_l0rt  +  3)2.(GZ/2  +  7^-20)3.(wi3_^^^3)^ 

Find  the  highest  common  measure  of:  . 

16.  fl* +  4^3  + 4^2^     a^b- 4:6(1),     a^h  +  ba^b-^Qa^h. 

17.  x'-{y  +  l)\    y'-{x  +  l)\     \-{xi-y)\ 

18.  a^-b\    (i'^-ab^b'',    a^  +  crb^  +  bK 

19.  x^  +  4:X^  +  4:(r  +  S,     a^-a^-x-2. 

20.  4a;*  +  9a;3  +  2a;2-2a;-4,     3.^-3  +  Sa;^  -  a:  +  2. 

21.  2x^+i2a~^)x^-{9a  +  6)x-\-27,     22;2_]^3^_,_I3^ 

22.  4a2-4«a;-15a;2,     6a2_|.;j.^^_3^.2^ 

23.  7ix^  +  3nx^i/  — 2)1X1/^  — 2)iy^,     4:mx^  +  mo^y  —  2mxy'^  —  ^my^, 

24.  a;*-joa:3+(g-l).T2+j?;3:-(7,     x^-qx^-\-{p-l)x^-Vqx-p. 

25.  a;3+(4«  +  ^,):c2+(3«2_^4«^.)a;  +  3^,2j^ 

a:3  +  (2rt-^)a:2_(3^^2^2«^)2;+3rt2J. 

26.  a^e^  +  e^-a^-l,    (f^-2  +  «-').(e^-2  +  e-^). 


'130 


MEASURES  AND  MULTIPLES. 


[IV,  PR 


Note  1.  The  arrangement  of  terms  may  be  as  to  the  rising 
powers  of  the  letter  of  arrangement,  or  as  to  the  falling  powers. 

E.g.,    2:i:^  +  lla;H20.T  +  21     and     a^-x-6; 
or         21+20a;  +  ll:?;2_{_2^-3    ^^^^     q  ^^^-a^. 

That  arrangement  is  commonly  best  which  makes  the  trial 
divisor  smallest;  and  at  any  step  of  the  work  the  highest  or 
the  lowest  term  of  the  divisor  may  be  used  as  trial  divisor. 
E.g.,  to  find  the  h.  c.  msr.  of  a^  +  Ssf +  5x  + 3,  x^ +  6x^  +  93! -i^i^ 


write   arH3a;2+   5a: +  3 
9x^  +  12x  +  3 

0?-  Gx'- 
-21x'-28x^- 

-7x 
-Ix 

22.tH222;2 

(22a;^ 

iC  +  1 

1 

ic3  +  3a;H5a:  +  3 

3-7a: 

^x^^r^x^-\ 

ZxW 

3:&\%x 

x-\-l 


(c)  The  'prime  factors  not  known ,  three  or  more  entire  num- 

bers :  find  the  highest  common  measure  of  any  tivo  oj 
them  (preferably  the  lowest),  the  highest  common  meas- 
ure of  this  measure  and  the  next  number,  and  so  on. 

E.g.,  to  find  the  highest  common  measure  of 

x^'-^x-U,     a;2-10rc  +  21,    x'-Gx^-lQx-^-S^i 

then*.*of    x^-^x  —  'i2,    ic^-10a:  +  31,    the  h.  c.  msr.  is    rr-S, 

and      x  —  3    measures    Q^  —  Qx^  —  lQx  +  Si, 

.',x  —  ^    is  the  measure  sought.  q.e.f. 

(d)  Some  or  all  of  the  numbers  fractions .  divide  the  highest 

common  measure  of  the  entire  numbers  and  the  numer- 
ators by  the  lowest  common  multiple  [pr.  5]  o/  the 
denominators.  [th.  16. 

E.g.,  to  find  the  highest  common  measure  of  the  fractions 

{x'  +  x-l2)/{x-b),     (a;2-10a;  +  21)/(.T  +  5): 
then  •.'  the  h.  c.  msr.  of  the  numerators  is  .t  — 3,  [above, 

and       tlie  1.  c.  mlt.  of  tlie  denominators  is  a;^  — 25,  [inspection. 
.-.  {x  —  'd)/{^^^  —  2b)  is  the  measure  sought. 


4,  §4]  THE  HIGHEST  COMMON   MEASURE.  131 


QUESTIOXS. 

Keduce   to  lowest   terms  by  means  of  the  highest  common 
measures  of  their  numeiators  and  denominalors; 

1-   :; -;:-. — ^ ^  •  ^» 


n^  +  x^  ^Lz£*  f^    xy-'^  +  l+j-^y 

a^  +  y-^  a^-yr^        ^  rc-^  +  ll^c-'-f  30 

"•     ^.7    ,    „-7*  *•     .J        ^,-7* 


Find  the  highest  common  measure  of; 

202,-«  -  12a;5+ 16:6*  - 15:^3  +  14:^-2  _  15^  ^  4^ 

10.  a:*-2;'-3x-2  +  5^-2,     :rS-2a:*-ar'  +  5r»-4a;  +  l. 

11.  '^x''+(ia-U)x-^ab-Va\  x^+ {2a-b):^-{2ab-a^)x- a^b 

12.  2x^  +  3^  +  1,     2x-2  +  5a:  +  2,     2a;3  +  5a;2_4^_3^ 

13.  12.r'-2^-7a:  +  2,     18^^-9a:2_8^_|.4^     36a,'*-25x-2  +  4. 

14.  3A  +  6«^-^  +  3^lc,     12rt2-12«5  +  3^^     lOa^  +  Sa^-Sft^. 

15.  x^tj  +  12.6-'^  -f-  35,r?/,     rt^  +  3.^'*  -  28.c'^,     2.-^2  -  aj^2  -  56./'2. 

16.  Prove  that  the  highest  common  measure  of  two  or  more 

numbers  is  the  reciprocal  of  the  lowest  common  mul- 
tiple of  their  recipi'ocals. 

Find  the  highest  common  measure  of: 

1-    01       ^2.      ^       ^       i«    ^\  ^^     ^     A     ^      x"     x"     X 
64     16     128  y^     y     y  2y    y     2     y 

af-^x      a-^-8a;f  16       c^-2x-8 
x^  +  y^*    x^  —  x^y-\y^^        x^^y^ 

6a:*fl0a;-24         22:*-2a:-24        8:^^22x-^ 


20. 

21. 


a:^-i-5^f  6a.  '     x'y  —  x^y  —  ^xy''        x^-y  —  ^y     * 
x^^2cx^&       x^-{-{n-h)T-bc  Z7?-Z(^ 


[X  ^a){:£-\  b)      x'  -\-{a  +  c)x  +  ac     x^+{b-^ c)x  +  bo 


132  MEASURES  AND  MULTIPLES.  [IV,  Pr 

§5.   THE   LOWEST   COMMON   MULTIPLE. 

PrOB.  5.    To  FIND  THE  LOWEST  COMMON  MULTIPLE    OF  TWO 
OE  MORE   NUMBERS. 

(a)  The  prime  factors  of  all  tlm  numbers  Joioiun :  miiUiply 

together  all  the  different  prime  factors,  each  with  the 
greatest  exponent  it  has  in  any  of  the  numbers,  [th.  14. 

E.g.,  of  ^a¥G-^,  12«^MS  15a*  +  21«W,  the  prime  factors, 
in  their  highest  powers,  are  2^,  3^,  a^,  h^,  6^,  d^,  ba^  +  '^bd, 

and  the  lowest  common  multiple  is  180rt*Z/^^*  +  252«^/^*(^^^ 

[b)  Tlie  prime  factors  not  known,  two  entire  numbers:  divide 

either  number  by  their  highest  common  measure  and 
multiply  the  quotient  by  the  other  number,  [th.  14  cr.  3. 
¥i.g.,  to  find  the  lowest  common  multiple  of 
x'  +  x-VZ,    a;'^-10.c  +  21: 

then  their  h.  c.  msr.  is     a:  — 3,     and  the  multiple  sought  is 
{x'  +  x-U)'.{,c-'6)-{x''-li)x  +  '^l)^u?-Qx'-l^x  +  S4:. 

{c)  The  prime  factors  not  known,  three  or  more  entire  mem- 
bers :  find  the  loicest  cotninnji  multiple  of  any  two  of  the 
Clumbers  (preferably  the  highest),  tlie  loivest  common 
multiple  of  this  imiltiple  and  tlie  next  number,  and  so  on. 

E.g.,  to  find  the  lowest  common  multiple  of 

a;HG.i-H9:i;  +  4,     x^-V  Zx^'-Ux-l'Z,     x^-lx-iy: 

then  the  1.  c.  mlt.  of  x^  +  Gx^  +  ^x  +  4,  ar"  +  1x^  -  11a;  - 12  is 
(x  +  \)\x  +  4.){x-^), 

and  the  1.  c.  mlt.  of     (x^-\f{x  +  4:){x-Z),     a?-lx-(y     is 
(:x  +  \)\x\-A){x-6)[x  +  2). 

(d)  Some  or  all  of  the  numbers  fractions :  divide  the  lowest 
co7nmon  muUi2)le  of  the  entire  numbers  and  numerators 
by  the  Jiighest  common  measure  of  the  denominators. 

E.g.,  to  find  the  lowest  common  multiple  of  the  fractions 
{x^  -  y'')/{a''  -a-  20),     (:f«  -  y')/{a^  +  Qa  +  8) : 

then  •.•  the  1.  c.  mlt.  of    x^  —  y^,     x^  —  y*    is    x^—y^, 

and       tlie  li.  c.  msr.  of     a'^  —  a  —  2{),     a^-{-^a  +  S     is     ^  +  4, 
.%  the  multiple  sought  is     0^'^  — y)/(«  +  4). 


e,S5]  THE   LOWEST   COMMON   MULTIPLE.  133 

QUESTIOIs'S. 

Find  the  lowest  common  multiple  of: 
L  6x^-x-l,     2.v^  +  3x-2.     2.  a  +  1,     1-a,    aH«  +  l. 
3.  a^-b\     a^  +  2ab  +  h\  4.  x^-Gx'^-x  +  dO,    a^ -\- 3x^ - '^. 

Reduce  to  their  lowest  common  denominators: 

x^  +  y^     ^  —  'f     X? -\- xy -\- if"     o^  —  xy^-y^     x  —  y      x-^y 
x^  —  y'^     x^  +  y'^'         x^  +  y^     '         x^  —  y^    '     x-\-y'    x  —  y 
aVb     a-h     a'  +  W      a'-W      a^  +  W      a^-¥ 
a-b'    a  +  b'    a^-W'    aF+^'    oF^''     '^Tb^' 
Add  the  fractions: 

^         1  3  5  2  3  2a:-3 

'•  2{a  +  xY     4(«-^)'     Q{ii'-x'y         '  x'     l-2x'     4.x'-i' 

10.  l/4:0^(:x  +  y),    l/2x\x^  +  y'),    \/4.x\x-y),    \j2x\:x?-f\ 
Find  the  lowest  common  multiple  of: 

11.  T^-^ci",    x^^^ax^'-Via^x-vM,     :^-2a3^-h4:a^x-8a\ 

12.  4:{x^-y'){x-y)\     \2(x'-y')(x-y)\     20(x''-yy, 

13.  a^-4:a%     x^  +  eax'^  +  Ua^'x  +  Sa^     x^-eax'^  +  Ua^x-SaK 

14.  7n^n^-2mu^-i-7i\     x^  +  2xy  +  y^,     mnx-nx-ny +  mny, 

15.  x^-y^,     ^-^^     ^-y^9     ^-2xy-Vy'^,     x'^  +  2xy  +  y\ 

16.  ahri'  +  Sam  +  lQ,     a^-4:,     a^  +  3a  +  2,     a^-da  +  2. 

17.  a^-6x^  +  Ux~6,     x^-dx^  +  26x-24:,     .r^-8a;2  +  19a;-12, 

18.  a'  +  a^^  +  b\     a'  +  3^^^^  +  ia'b''  +  3^^*^  +  bK 

19.  4a«-4a*-29«2_21,     4rtH24rt*  + 41^^21. 
a^-\-x      a?  —  x         x^'^-x  x^  —  x 


20. 
21. 
22. 
23. 


2-2rc2'     8  +  8?     12  +  12.1-2'     l-^2x  +  2x'  +  x^' 

a^-h5x-h4:      2^2^-8      a:^  +  7.-g  +  12       .^'2-3.r-4 

x^-4:X  +  3'     'x^-x-VZ'      x^  +  x^2()'     x"  +  2x  -15' 

x^-if-z^-^2yz  x'-y^  +  z'-2xz 

x'  +  2xy+y^-  z''     -  -  3x'2  +  'dy'  -  Qyz  +  3?* 

2x^  +  x^-bx-?>       2x^bx^-x-Q  a^-\ 

a:*  +  2a^+~2^?         i«;2-2:?;-l    ''    x^-af-2x-l' 


134  MEASURES  AND  MULTIPLES.  [IV. 

§  6.  QUESTIONS  FOR  REVIEW. 

Define  and  illustrate: 

1.  An  integer;  an  entire  function  of  one  letter;  an  entire 
number. 

2.  A  multiple  of  an  integer;  a  measure. 

3.  A  multiple  of  an  entire  function  of  a  letter;  a  measure. 

4.  An  entire  factor  of  an  entire  number;  a  linear  factor. 

5.  A  common  multiple  of  two  integers;  a  common  measure. 

6.  A  common  multiple  of  two  entire  functions  of  a  letter; 
a  common  measure. 

7.  The  lowest  common  multiple  of  two  or  more  entire  num- 
bers; the  highest  common  measure. 

8.  A  prime  integer;  a  prime  entire  function  of  one  letter. 

9.  Two  numbers  prime  to  each  other;  a  composite  number. 

10.  State  the  axioms  that  relate  to  the  sum,  difference,  and 
product  of  two  entire  numbers. 

11.  Show  how  a  common  measure  of  two  entire  numbers  is 
related  to  their  sum;  to  their  difference;  to  the  sum  and  the 
difference  of  any  multiples  of  them. 

12.  State  Euclid's  process  for  finding  the  highest  common 
measure  of  two  entire  numbers,  with  proofs  and  illustrations. 

Show  that  every  remainder  so  found  is  the  difference  of  two 
multiples  of  the  given  numbers;  and  that,  if  the  numbers  be 
prime  to  each  other,  two  multiples  of  them  can  be  found 
whose  difference  is  either  a  unit  or  some  expression  that  is 
free  from  the  letter  of  arrangement. 

State  and  prove  the  converse  of  this  proposition. 

Prove  that  these  statements  are  true: 

13.  If  an  entire  number  be  prime  to  two  or  more  such  num- 
bers, it  is  prime  to  their  product. 

14.  If  two  entire  numbers  be  prime  to  each  other,  so  are 
any  positive  integer  powers  of  them. 

15.  If  an  entire  number  measure  the  product  of  two  such 
numbers  and  be  prime  to  one  of  them,  it  measures  the  other. 


S6]  QUESTIONS  FOR  REVIEW.  1^5 

16.  If  an  entire  prime  number  measure  the  product  of  two 
or  more  entire  numbers,  it  measures  at  least  one  of  them. 

17.  If  an  entire  prime  number  measure  a  positive  integer 
power  of  an  entire  number,  it  measures  the  number,  and  if 
the  number,  then  the  power. 

18.  If  an  entire  number  be  measured  by  two  such  numbers 
that  are  prime  to  each  other,  it  is  measured  by  their  product. 

19.  An  entire  number  can  be  resolved  into  prime  factors  in 
but  one  way. 

20.  A  common  measure  of  two  or  more  entire  numbers  can 
contain  no  factor  that  is  not  in  all  of  tiiem. 

21.  If  /a-,  an  entire  function  of  .r,  be  divided  by  x  —  a^  the 
remainder  is/^;  and  if /«  =  0,  a:  — «  is  a  measure  of  /a;. 

22.  If  the  numerator  and  denominator  of  a  fraction  be 
prime  to  each  other,  the  fraction  is  in  its  lowest  terms;  and 
so  is  every  integer  powder  of  it. 

23.  Which  factors  of  two  entire  numbers  are  factors  of  their 
highest  common  measure  ?  of  their  lowest  common  multiple  ? 

What  effect  has  it  upon  their  highest  common  measure  to 
multiply  or  divide  either  of  the  numbers  by  a  number  that  is 
not  a  factor  of  the  other  number  ?  by  a  number  that  is  a  fac- 
tor of  the  other  ?  upon  the  lowest  common  multiple  ?  to 
multiply  or  divide  both  numbers  by  the  same  number? 

24.  What  is  the  product  of  the  highest  common  measure 
and  the  lowest  common  multiple  of  two  numbers? 

Give  the  general  rule,  with  reasons  and  illustrations,  for: 

25.  Factoring  an  integer;  factoring  a  polynomial  of  known 
type-form;  factoring  an  entire  function  of  one  letter;  finding 
the  linear  factors  of  an  entire  function  of  one  letter. 

26.  Finding  the  highest  common  measure  and  the  lowest 
common  multiple  of  two  or  more  entire  numbers  whose  prime 
factors  are  known;  of  tw^o entire  numbers  whose  prime  factors 
are  not  known;  of  three  or  more  entire  numbers  whose  prime 
factors  are  not  known;  of  fractions. 


136  VARIATION,  PROPORTION,  INEQUALITIES.  [V, 

V.  VARIATION,  PROPORTION,  INEQUALITIES,  AND 
INCOMMENSURABLE  NUMBERS. 


Hitherto  concrete  numbers,  integers  or  fractions,  have  been 
found  by  counting  like  entire  units, — apples,  horses,  guests — 
or  by  measuring  by  some  definite  unit,  or  part  of  a  unit,  and 
counting  the  number  of  times  the  unit  is  contained  in  the 
thing  measured.  Abstnict  units  and  simple  fractions  express 
repetitions  and  partitions  and  their  combinations. 

Such  numbers  are  coinmensnrcible  niimhers. 

Hitherto  also  a  number  has  been  thought  of  as  something 
fixed  and  definite,  and  if  a  letter  were  used  to  denote  a  num- 
ber, it  was  some  fixed  and  definite  number. 

Such  numbers  are  constants. 

But  now  come  two  new  notions:  that  of  changing  vakieSj 
variables;  and  that  of  numbers,  definite  and  distinct,  which 
however  cannot  be  expressed  by  repetitions  and  partitions, 
incommensurable  niimhcrs. 

§  1.    VARIATION. 

If  a  boy  count  apples — one,  two,  three-  •  •,  as  he  drops  them 
into  a  basket,  the  number  of  apples  in  the  basket  changes  and 
increases;  or,  if  he  have  twelve  apples  at  the  start  and  take 
them  out  one  by  one,  the  number  left  changes  and  decreases — 
twelve,  eleven,  ten,-  •  •,  and  the  number  of  apples  in  the  basket 
is  a  variable.  Or,  if  he  count  the  horses  that  pass  through  a 
gate  into  a  field — one,  two,  three,-  •  •,  or  the  guests  as  they 
rise  from  table,  or  the  3's  he  gets  when  he  reduces  the  simple 
fraction  1/3  to  a  decimal:  in  all  these  cases  the  numbers  so 
found  are  variables. 

So,  if  the  cross-section  of  the  trunk  of  a  growing  tree  be  a 
circle,  it  is  a  circle  whose  radius,  circumference,  and  area  are 
all  variables;  and  a  growing  peach  is  a  sphere  wliose  radius, 
circumference,  surface,  and  volume  are  variables. 

So,  with  a  sum  of  money  at  interest,  the  principal  and  rate 
are  constants;  the  time  and  accrued  interest  are  variables. 


§1]  VARIATION.  137 

QUESTIONS. 

1.  Show  that  a  variable  may  increase  or  decrease  by  regular 
additions  or  subtractions,  or  by  irregular  ones. 

2.  In  annexing  successive  3's  to  the  decimal  expression  of 
the  fraction  1/3,  are  the  successive  additions  to. the  value  of 
the  variable  the  same  or  different  ? 

3.  Is  the  number  1/9  a  variable?  is  its  decimal  expression 
a  variable  ?  can  the  same  number,  then,  be  both  a  variable 
and  a  constant,  or  is  1/9  not  equal  to  its  decimal  expression  ? 

4.  Is  the  reciprocal  of  a  variable  a  constant  or  a  variable  ? 
How  does  the  reciprocal  of  an  increasing  variable  change  ? 

the  opposite  ?  the  opposite  of  the  reciprocal  ? 

5.  In  general,  is  the  sum  of  two  variables  a  constant  or  a 
variable?  their  difference?  their  product?  their  quotient  ? 

Show  by  examples  that  the  sum,  the  difference,  the  prod- 
uct, and  the  quotient,  of  two  variables  may  be  constants. 

6.  In  the  case  of  an  express  train  running  over  a  long,  level, 
straight  track,  with  the  same  pressure  of  steam,  which  of  the 
following  elements  are  variables:  the  time  since  the  train 
started,  the  distance  it  has  run,  the  speed,  the  relation  of  the 
distance  to  the  speed,  its  relation  to  the  time  ? 

7.  If  the  wine  in  a  full  cask  run  into  an  empty  one,  what 
two  variables  are  there  during  the  process?  what  constant? 

Do  these  two  variables  vary  in  the  same  way  ? 

8.  The  diagonal  of  a  square  whose  side  is  a  is  |/2«^;  what 
is  the  diagonal  of  a  square  whose  side  is  ^  ? 

What  relation  does  the  diagonal  of  a  square  bear  to  a  side? 
If  the  length  of  a  side  change,  does  the  diagonal  change  ? 
Does  the  relation  between  the  side  and  diagonal  change? 

9.  "What  is  the  area  of  a  square  whose  side  is  a  ?  of  one 
whose  side  is  Z*?    What  is  the  relation  of  the  area  to  the  side  ? 

Is  the  area  of  a  growing  square  a  fixed  number  of  times  the 
side  ?     What  definite  law  connects  these  two  magnitudes? 

If  the  length  of  a  side  can  not  be  exactly  expressed,  what 
effect  has  that  on  this  law  ? 


138  VARIATION,  PROPORTION,  INEQUALITIES.  [V, 

CONTINUOUS   AND   DISCONTINUOUS   VARIABLES. 

If  a  concrete  variable,  in  passing  from  one  value  to  another, 
pass  through  every  intermediate  value,  it  is  a  conti7iiious 
variable;  otherwise  it  is  a  discontinuous  variable. 

So,  the  abstract  ratios  of  these  concrete  variables  to  the 
constant  measuring  unit  are  continuous  or  discontinuous 
variables. 

E.g.,  if  a  man  has  waited  two  hours  for  an  incoming  steamer, 
he  has  also  waited  an  hour,  an  hour  and  a  quarter,  an  hour 
and  a  half,  and  every  other  portion  of  time  less  than  two  hours 
that  can  be  named  or  conceived  of;  and  he  has  been  conscious 
of  the  continuous  passage  of  time. 

So,  if  he  run  along  the  street  he  knows  that  he  cannot  f^et 
from  one  fixed  point  to  another  without  going  through  every 
intermediate  point  of  some  path,  and  that  the  distance  run 
has  a  continuous  growth. 

But,  of  the  regular  polygons  inscribed  in  a  circle,  one  may 
have  three  sides,  another  four,  another  five,  and  so  on;  but 
no  one  can  have  four  and  a  half  sides,  nor  five  and  a  quarter 
sides.  The  number  of  sides,  the  perimeter,  and  the  area  are 
all  discontinuous  variables. 

RELATED    VARIABLES. 

If  two  variables  be  so  related  that  the  value  of  one  of  them 
depends  upon  that  of  the  other,  the  first  variable  is^  fund io?i 
of  the  other.  [compare  IV,  §  2. 

E.g.,  with  a  given  principal  and  rate,  the  accrued  interest  is 
a  function  of  the  time. 

So,  the  circumference  and  area  of  a  growing  cii-cle  are 
functions  of  the  radius. 

So,  of  the  regular  polygons  inscribed  in  a  given  circle,  the 
perimeter  and  area  are  functions  of  the  number  of  sides. 

A  variable  may  be  a  function  of  two  or  more  variables. 

E.g.,  the  volume  of  a  stick  of  timber  is  a  function  of  its 
length,  breadth  and  thickness;  and  the  cost  is  a  function  of  the 
length,  breadth,  thickness,  and  cost  per  foot. 


«1]  VARIATION.  139 

QUESTIONS. 

1.  Can  a  variable  be  continuous,  and  its  reciprocal  be  dis- 
continuous ?  its  opposite  ?  its  square  ?  its  square  root  ? 

2.  State  whether  the  numbers  below  are  constants  or  vari- 
ables, and  if  variables,  whether  continuous  or  discontinuous: 

the  length  of  a  line  revolving  about  the  centre  of  a  circle  as 
a  pivot  and  reaching  to  the  circumference; 

such  a  line  revolving  about  any  other  point  than  the  centre; 
the  principal  at  simple  interest;  at  compound  interest; 
the  size  of  an  angle  if  the  bounding  lines  be  lengthened; 
the  number  of  telegraph  poles  passed  by  the  electric  current. 

3.  If  a  falling  body  has  at  one  moment  a  velocity  of  30  ft. 
a  second,  and  later  a  velocity  of  40  ft. ;  how  many  other  dif- 
ferent velocities  has  it  had  between  these  times  ? 

4.  What  relation  does  the  perimeter  of  a  square  bear  to  the 
length  of  one  side  ?  If  the  side,  for  any  reason,  can  not  be 
measured,  does  that  fact  affect  this  relation  ? 

5.  The  circumference  of  a  circle  is  readily  seen  to  be  some- 
what more  than  three  times  the  diameter;  the  exact  relation 
is  found  by  geometry,  and  it  is  usually  represented  by  the 
(xreek  letter  n^  read  pie,  whose  value  is  nearly  3.1416:  what 
is  the  relation  of  the  circumference  of  a  circle  to  its  radius  f 
does  this  relation  depend  in  any  way  on  the  length  of  the 
radius  ?  on  the  length  of  the  circumference  ? 

6.  If  the  radius  of  a  circle  increase,  does  the  circumference 
increase  at  the  same  rate  or  at  a  greater  or  less  rate  ?  the  area  t 

7.  As  interest  is  computed  in  business,  for  days  and  not  for 
fractions  of  a  day,  thus  making  the  time  discontinuous,  is  the 
accrued  interest  a  continuous  or  a  discontinuous  function  of 
the  time  ?  the  amount  ? 

8.  If  a;  be  a  continuous  variable,  is  a^  —  2a^  +  x-l-D  a  con- 
stant, or  a  continuous,  or  a  discontinuous,  function  of  x?  1/x? 

9.  If  the  diameter  of  a  peach  grow  continuously,  does  the 
circumference  grow  continuously?  the  surface?  the  volume? 
the  weight?  the  value? 


140  VARIATION,  PROPORTION,  INEQUALITIES.  [V, 

EELATIVE   YARIATION". 

One  number  varies  directly  as  another  if  when  the  second 
is  doubled  so  is  the  first,  when  the  second  is  tripled  so  is  the 
first,  when  the  second  is  halved,  so  is  the  first,  and  so  on. 
Bi.g,,  with  a  constant  principal  and  rate  the  interest  accrued 

varies  directly  as  the  time. 
So,  with  men  of  equal  efficiency,  the  amount  of  work  done  in 
a  day  varies  directly  as  the  number  of  men  employed. 
The  sign  of  variation  is  a,  read  varies  as. 
E.g.,  if  i  stand  for  simple  interest  and  t  for  time,  then  ioat. 
So,  if  w  stand  for  the  work  done  in  a  daj  and  tn  for  the  num- 
ber of  men,  then  wocvi. 

If  two  abstract  numbers,  or  two  concrete  numbers  of  the 
same  kind,  vary  in  such  wise  that  some  relation  between  their 
magnitudes  shall  continue  to  hold  true,  this  relation  may  be 
expressed  by  an  equation. 

E.g.,  if  d  be  the  diameter  of  a  circle  and  r  its  radius; 
then     doer,     and     d=2r. 

One  number  varies  inversely  as  another,  if  when  the  second 
is  doubled  the  first  is  halved,  when  the  second  is  tripled  the 
first  is  trisected,  when  the  second  is  halved,  the  first  is  doubled, 
and  so  on.  If  the  numbers  be  abstract  one  varies  directly  as 
the  reciprocal  of  the  other. 
E.g.,  the  time  of  running  a  fixed  distance  varies  inversely  as 

the  speed. 
So,  the  illuminating  power  of  a  light  varies  inversely  as  the 

square  of  its  distance. 

So,  if  Xf  ?/,  be  two  abstract  ni;mbers  and  x  vary  inversely  as  y\ 
then     xccX/y,     and     x  —  k/y,  wherein  h  is  some  constant. 

One  number  varies  jointly  as  two  others  if  it  vary  directly 
as  their  product. 

E.g.,  if  w  stand  for  the  wages  earned,  m  for  the  number  of 
men  and  t  for  the  time,  then   wocm-t,   or   w  =  k-ni-t, 

[k,  a  constant. 


§1]  VARIATION.  141 

A  number  may  vary  directly  as  one  number  and  inversely  as 
another. 
E.g.,  the  time  of  running  varies  directly  as  the  distance  run 

and  inversely  as  the  speed. 
So,  the  value  of  a  fraction  varies  directly  as  the  numerator 

and  inversely  as  the  denomiuiitor. 

QUESTIONS. 

1.  What  is  meant  by  saying  that  the  circumference  of  a 
circle  varies  as  the  diameter?  that  the  radius  varies  as  the 
circumference  ?  that  the  area  varies  as  the  .square  of  the  radius? 

If  a  stone  be  di-opped  into  a  pond  of  water,  how  many  of 
these  relations  will  be  true  at  each  and  every  stage  of  the 
growing  circular  wave  ? 

2.  Does  the  light  received  from  a  lamp  vary  as  its  distance? 
how  then  ?  the  entire  cost  as  the  number  of  like  articles  bought? 
the  time  needed  for  a  piece  of  work  as  the  number  of  men? 

3.  If  a  lady  spend  the  same  sum  for  gloves  every  year,  but 
the  price  increase,  how  does  the  number  of  pairs  bought  vary  ? 

4.  Of  rectangles  having  the  same  area,  how  does  the  width 
vary?  the  length  ?  the  perimeter?  the  diagonal? 

5.  If  the  sum  of  two  numbers  be  constant,  will  one  vary  as 
the  other  ?  inversely  as  the  other  ?  if  the  product  be  constant? 
the  quotient  ?  the  difference  ? 

6.  If  at  different  times  varying  numbers  of  like  things  be 
bought,  and  at  varying  prices,  how  do  the  bills  vary  ? 

7.  If  the  dividend  be  constant  and  the  divisor  vary,  how 
does  the  quotient  vary?  if  the  divisor  be  constant  and  the 
dividend  vary?  if  the  dividend  and  divisor  both  vary? 

May  the  dividend  and  divisor  both  vary  and  the  quotient 
remain  constant? 

8.  In  simple  interest  is  the  principal  a  constant  or  a  varia- 
ble ?  with  a  fixed  principal  and  rate,  how  do  the  interest  and 
time  vary  ?  that  a  fixed  principal,  at  interest  for  various 
periods  of  time,  may  bring  in  the  same  interest  for  each 
period,  how  must  the  rate  be  related  to  the  time  ? 


142  VARIATION,  PROPORTION,  INEQUALITIES.  [V, 

§2.   PROPORTION. 

The  ratio  of  one  quantity  to  another  quantity  of  the  same 
kind  is  the  number  found  by  measuring  the  first  quantity  by 
the  other  as  a  unit;  i.e.,  it  is  the  multiplier  that,  acting  on 
the  second  quantity  as  a  unit,  gives  the  first  quantity  as  result. 

The  ratio  of  two  like  numbers  is  the  quotient  of  the  first  by 
the  other,  and  it  is  an  abstract  number. 

E.g.,  the  ratio  of  12  bushels  of  wheat  to  4  bushels  of  wheat  is 
3,  and  the  ratio  of  4  bushels  to  12  bushels  is  1/3. 

The  ratio  of  «  to  J  is  written     a:b    or     a/b. 

No  ratio  is  possible  between  unlike  numbers. 

The  ratio  a:l)  is  the  direct  ratio  of  a  to  b,  and  b'.  a  is 
the  inverse  ratio. 

In  a  direct  ratio,  the  dividend  is  the  antecedent  and  the 
divisor  is  the  consequent. 

The  equation  of  two  ratios  is  a  jp'roportion,  and  the  four 
numbers  forming  the  two  ratios  are  the  terms  of  the  propor- 
tion, or  the  io\x\:  projmrtionals. 

E.g.,  if  the  ratio  of  a  to  b  equals  that  of  c  to  d,  this  fact  may 
be  stated  in  a  proportion  in  three  ways: 
a-.bwc'.d,    a/b  —  c/d,    a\b  —  c\d. 

The  first  form  is  read,  a  is  to  b  as  c  is  to  d;  and  the  others 
more  briefly,     a  to  b  equals  c  to  d. 

A  proportion  between  concrete  numbers  is  but  an  expanded 
statement  of  direct  variation. 
E.g.,  2  years  :  3  years  =  2  years'  interest  :  3  years'  interest 

means  that,  the  principal  and  rate  being  constant,  tlie 

interest  varies  as  the  time. 

The  first  and  last  terms  of  a  proportion  are  the  extremes, 
the  second  and  third  the  means,  the  first  and  third  the  ante- 
cedents, the  second  and  fourth  the  consequents. 

If  the  means  of  a  proportion  be  the  same  number,  the  com- 
mon term  is  the  meaii  ijroportioiud,  and  the  last  term  is  the 
third  jJrojjortionaL 


82]  PROPORTION.  143 

E.g.,  in  the  proportion  a:h  =  h'.c,  ^  is  the  mean  propor- 
tional between  a,  c,  and  c  the  third  proportional  to  a,  t. 
If  three  or  more  ratios  be  equal,  they  may  be  written  in 
succession,  with  the  sign  =  between  them.  Such  an  expres- 
sion is  a  continued  j^TOportion, 
E.g.,     a'.h  —  c\d=e\f—g\li, 

QUESTION'S. 

1.  Can  a  ratio  be  concrete  ?  can  it  be  negative  ? 

2.  If  two  numbers  be  equal,  what  is  their  ratio  ?  opposites  ? 
if  the  antecedent  be  the  larger  ?  the  consequent? 

3.  AVhat  two  relations  between  the  antecedent  and  conse- 
quent make  the  direct  and  inverse  ratios  equal  ? 

4.  If  a  ratio  be  zero,  which  term  is  zero?  if  infinite  ? 

5.  Name  two  other  numbers  that  have  to  each  other  the 
same  ratio  as  10  has  to  5,  and  write  these  four  numbers  as  a 
proportion.  Find  other  pairs  of  numbers  having  the  same 
ratio,  and  of  them  all  make  a  continued  proportion. 

6.  As  a  statement  of  variation,  interpret  the  proportion 

2  hrs.  :5  hrs.  =  distance  run  in  2  hrs. :  distance  run  in  5  hrs. 

7.  As  a  proportion  write  the  statement  that  the  areas  of 
circles  vary  as  the  squares  of  their  radii. 

8.  If  the  antecedents  of  a  proportion  be  equal,  what  is  true 
of  the  consequents?  if  one  antecedent  be  ten-fold  the  other? 

9.  If  the  first  antecedent  be  the  square  of  its  consequent,  is 
the  second  antecedent  likewise  the  square  of  its  consequent  ? 

10.  Of  the  continued  proportion  at  the  top  of  the  page,  make 
six  different  simple  proportions. 

11.  What  effect  is  produced  on  a  ratio  by  doubling  the 
antecedent  ?  the  consequent  ?  both  terms  ? 

12.  Regarding  a  proportion  as  an  equation  between  two 
fractions,  show  why  it  is  allowable  to  multiply  or  divide  both 
antecedents  by  the  same  number  ?  both  consequents  ?  both 
terms  of  one  ratio?  all  four  terms?  to  multiply  one  antece^ 
dent  and  divide  the  other  consequent  by  the  same  number  ? 


144  VARIATION,  PROPORTION,  INEQUALITIES.        [V,Th& 

PROPERTIES   OF   PROPORTIOJs^S. 

A  proportion  is  but  an  equation  whose  members  are  frac- 
tions; and  the  principles  already  established  for  fractions  aad 
for  equations  apply  directly  to  the  proportions  used  in  the 
proof  of  the  theorems  below. 

In  these  proofs,  the  proportionals  are  abstract  numbers. 

Theor.  1.  In  any  proportion,  the  product  of  the  extremes 
equals  that  of  the  means. 
Let      a\l  =  c:d,    then  will    ad=  he. 

For  •.*  a/b  =  c/d,  [hyp. 

/.  ad = be,  Q.  E. D.         [mult,  by  hd:  ax.  mult. 

Cor.  1.  Either  extreme  is  the  quotient  of  the  product  of  the 
means  luhen  divided  by  the  other  extreme;  and  so  for  the  means. 

Cor.  2.  A  mean  proportional  is  the  square  root  of  the  prod- 
uct of  the  extremes, 

Theor.  2.  If  the  'product  of  two  mimhers  equal  the  product 
of  two  others,  the  four  numbers  may  form  a  projyortion,  in 
iMch  the  factors  of  one  product  shall  be  the  extremes  and  those 
of  the  other  product  the  means. 
Let  ad— be;  then  a:b  =  c:dy  a\c=b\d. 
For  the  first  proportion,  divide  both  members  of  the  equation 
ad— be    by  bd\  for  the  other,  divide  by  cd, 

Theor.  3.  In  any  proportion,  the  two  means  may  change 
places.  [alternation. 

Let    a\b  —  c\d\    then    a:c=h',d. 

Prove  by  aid  of  theors.  1,  2. 

Theor.  4.  In  any  proportion,  the  consequents  may  change 
palaces  with  their  antecedents,  [inversion. 

Let    a:b  =  c:d;     then     h:a  —  d:c. 

Prove  by  aid  of  theors.  1,  2. 

Theor.  5.  In  any  proportion  the  sum  of  the  first  two  terms 
is  to  the  first  term  or  the  second  as  the  sum  of  the  last  two  terms 
is  to  the  third  term  or  the  fourth,  [composition. 


1.  2, 3.  4,  5,  §  2]  PROPORTION.  145 

QUESTIONS. 

1.  "What  is  the  test  of  the  correctness  of  a  given  proportion  ? 

2.  Supply  the  missing  terms  in  the  proportions  below: 

6:4=^9:      ;  7;     =-35:-.5;  -9:     =     :16; 

:3=zO:10;  :16  =  4:     ;  :-9  =  16:   . 

Is  there  more  than  one  way  of  completing  the  third  of  these 
proportions  ?  how  many  ways?  could  the  two  means  be  equal  ? 

3.  How  is  theor.  2  related  to  theor.  1  ? 

4.  In  proving  theor.  3,  what  multiplier  will  change  the  given 
ratio  a/h  to  the  desired  one,  a/c  ? 

5.  How  is  the  reciprocal  of  a  number  found  ?   Prove  theor.  4. 

6.  Clear  the  equation  a/b  =  c/d  of  fractions,  and  see  what 
multiplier  will  make  the  first  member  d/c;  and  so  make  a  new 
proof  of  theor.  4. 

7.  If  the  extremes  of  a  proportion  be  to  each  other  as  the 
means,  the  ratios  are  each  unity. 

8.  If  two  terms  of  one  proportion  be  the  same  as  the  two 
like  terms  of  another,  the  four  terms  that  are  left  may  form 
a  proportion. 

9.  The  direct  ratio  of  two  numbers  is  the  inverse  ratio  of 
their  reciprocals. 

10.  May  the  reciprocals  of  any  four  proportionals  form  a 
proportion  ?  their  opposites  ?  the  opposites  of  their  reciprocals  ? 

11.  May  any  four  proportionals  be  written  in  reverse  order? 
their  opposites  ?  the  ojiposites  of  their  reciprocals  ? 

12.  If   m\a  =n\h  and  c   :m  =  d  :  n,     then    a:i  =  c:d. 
So,  if  a  :m  =  b:n  and  c  :  d  =  m: n, 

13.  If  a  :m  =  7i:b  and  c  :m  =  7i:d,  then  « :  1/^  =  c :  1/fZ 
So,  if  a  1771  =  71  :b  and  m:c  =d:7i. 

14.  No  single  number  can  be  added  to  each  of  the  proper- 
tionals  a,  b,  c,  d  and  leave  the  sums  in  proportion. 

15.  If  there  be  a  single  number,  not  0,  that  may  be  added 
to  each  antecedent  without  destroying  the  proportion,  what 
relation  have  the  terms  of  the  proportion? 


146      INEQUALITIES,  INCOMMENSURABLE  NUMBERS.    [V,  Ths 

Theor.  6.  In  any  pro2)ortion  the  difference  of  tlie  first  two 
terms  is  to  the  first  term  or  the  second  as  the  difference  of  the 
last  two  terms  is  to  the  tliird  term  or  the  fourth.  [division. 

Cor.  1.  The  sum  of  the  first  tioo  terms  is  to  their  difference 
as  the  sum  of  the  last  two  terms  is  to  their  difference. 

Cor.  2.  The  snm  of  the  antecedents  is  to  their  difference  as 
the  sum  of  the  consequents  is  to  their  difference. 

Theor.  7.  If  two  proportio7is  be  multiplied  together,  or  if 
one  he  divided  hy  the  other,  term  hy  term,  the  results  are  pro- 
portional. 

Theor.  8.  Lihe  poioers  and  lihe  roots  of  the  terms  of  a  p^ro- 
portion  are  proportional. 

Theor.  9.  In  a  continued  proportion,  the  sum  of  all  the 
antecedents  is  to  the  sum  of  all  the  consequents  as  any  antece- 
dent  is  to  its  consequent, 
¥oY,\eta:b  —  c:d  —  e:f=-" 
then  *.•  a /a  -  h/b,     c/a  -  d/b,     e/a  =f/b,  •  •  • 

/.  (a  +  c  +  e+  "  '):a=(bi-d-i-f+  "  '):b,  [add. 

.•.  {a-\-c  +  e-\ ):  {b  +  d-i-f-\ )  =  a:b.     [alternation. 

Of  these  nine  theorems,  only  theors.  4,  5  are  always  directly 
applicable  to  concrete  numbers;  if,  however,  all  the  terms  be 
of  the  same  kind,  theors.  3,  9  also  apply. 

Moreover,  though  the  terms  of  a  proportion  be  all  concrete, 
their  ratios  are  abstract,  and  so  are  the  products,  quotients, 
powers  and  roots  of  these  ratios;  and  these  results  may  be  used 
as  operators  on  concrete  units. 

E.g.,  if  a  days :  b  days  =  $c :  $d,  and  e  men :/  men  =  $g :  th, 
then  ae  days'  labor:  bf  days'  labor  =  $c'_^:  $dh. 
For  the  proportions  give  the  abstract  equations 

a/b  =  c/d,  e/f=g/h,  ae/bf—cg/dh, 
and  •.•  ae  days'  labor  :  />/' days'  labor  =  r/6':  bf, 
and      %cg :  %dh  —  eg :  dh, 

,\  ae  days*  labor:  bf  days'  labor  =  $r^:  $dh. 


6,  7,  8,  9,  §  2  ]  PROPORTION.  14'? 

QUESTIOXS. 

1.  To  each  side  of  the  equation  ajh  —  c/d  add  1,  and  re- 
duce these  two  mixed  numbers  to  fractions;  from  the  resulting 
equation  find  a  hint  for  the  proof  of  theor.  6. 

Write  the  proportion  by  inversion,  then  prove  theor.  5. 

2.  Divide  the  equation  (a-\-V)/a  —  {c^-d)lc  by  the  equation 
(a  —  l))/a  =  {c  —  d)/c,  member  by  member,  and  prove  theor.  6 
cor.  1. 

So,  from  the  equation  a/'b  =  c/d  get  a/c=h/d,  and 
prove  theor.  6  cor.  2. 

3.  One  fraction  can  be  divided  by  another  by  dividing  the 
numerator  and  denominator  of  the  first  by  the  like  terms  of 
the  other;  and  such  a  process  is  equivalent  to  the  usual  one 
of  multiplying  by  the  divisor  inverted;  why  is  the  latter  rule 
oftener  used  than  the  other? 

Apply  this  method  in  proving  theor.  7. 

4.  If  the  first  two  terms  of  a  proportion  be  squared,  by 
what  is  the  first  side  of  the  equation  multiplied  ?  if  the  other 
two  be  also  squared,  is  the  same  multiplier  used  or  a  different 
one  ?    Hence  prove  theor.  8. 

5.  Why  may  the  two  antecedents  of  a  proportion  be  multi- 
plied by  one  number  and  the  two  consequents  by  another  ? 

6.  Show  why  theors.  1, 7, 8  are  of  no  direct  use  when  all  the 
terms  are  concrete  numbers;  and  why  theor.  2,  but  not  its 
converse,  may  apply  to  concrete  numbers. 

l,li{a->rh^-c  +  d)(a-'b-c  +  d)  =  {a-h  +  c-d){a-\-h-c-d), 
then    a,  h,  c,  d    are  proportionals. 

8.  If    a:b  =  c\d=e'.ff    and  h,  h,  I  be  any  numbers,  then 

a'.h={Jia-{-kc  +  le)\(Jih-^'kd-\-lf), 
and      «" :  If  =  {ha''  +  ^c"  +  /e") :  (W  +  kd""  +  //"). 

9.  The  distance  fallen  varies  as  the  square  of  the  time;  a 
body  falls  16  feet  in  one  second:  how  far  does  it  fall  in  two 
seconds?  in  three  seconds?  in  the  third  second?  in  five 
seconds?  in  the  last  two  of  the  five  seconds?  How  high  is  a 
tower  from  whose  top  a  stone  falls  in  3f  seconds  ? 


148     INEQUALITIES,   INCOMMENSURABLE  NUMBERS.  [v, 

§3.    INEQUALITIES. 

LARGER-SMALLER    INEQUALITIES. 

One  concrete  nnmber  is  larger  than  another  of  the  same 
kind  if  it  contain  more  units  than  the  otlier;  and  smaller  if 
it  contain  fewer  units.  The  positive  or  negative  quality  of  the 
numbers  is  not  thought  of,  but  only  their  magnitudes. 

One  abstract  number  is  larger  than  another  if  it  give  a  larger 
result  when  acting  on  the  same  unit. 

E.g.,  if  A  have  $50  and  owe  |30,  and  B  have  $G0  and  owe  $80; 
then  a's  assets  are  smaller  than  b's,  and  so  are  his  debts, 

a's  assets  are  larger  than  his  debts,  and  b's  are  smaller, 
a's  net  assets  are  as  large  as  b's  net  debts: 
and    +50<+60;     -30<-80;     +50>-30;     +60<-80. 

AXIOMS. 

1.  //'  of  three  numhers  the  first  he  larger  than  the  secondy  and 
the  second  be  equal  to  or  larger  than  the  third,  then  is  the  first 
7iumher  larger  than  the  third. 

3.  If  one  number  be  larger  than  another ,  and  if  each  of  them 
he  multiplied  by  the  same  number  or  by  equal  numbers,  then  is 
the  first  product  larger  than  the  other. 

3.  If  one  number  be  larger  than  another,  and  if  each  of  them 
be  divided  by  the  same  number  or  by  equal  numbers,  then  is  the 
first  quotient  larger  than  the  other. 

4.  If  one  number  be  larger  than  another,  and  if  the  same 
number  or  equal  numbers  be  divided  by  each  of  them,  then  is 
the  first  quotient  smaller  than  the  other. 

5.  If  one  set  of  numbers  be  larger  than  another  set  of  as 
many  more,  each  than  each,  then  is  the  j^voduct  of  the  first  set 
larger  than  the  product  of  the  others. 

6.  If  one  number  be  larger  than  another,  and  if  like  positive 
powers  or  roots  of  them  be  taken,  ^  en  is  the  poiver  or  root  of 
the  first  larger  than  that  of  the  other. 

7.  If  one  numher  be  larger  than  another,  and  if  like  nega- 
tive powers  or  roots  of  them  be  taken,  then  is  the  power  or  root 
of  the  first  smaller  than  that  of  the  other. 


§3]  INEQUALITIES.  149 

Not  all  of  these  axioms  are  axioms  in  the  sense  of  truths 
too  elementary  to  admit  of  proof,  for  some  are  directly  deriv- 
able from  others,  but  they  are  all  self-evident. 

QUESTIONS. 

1.  When  a  number  is  made  lai-ger,  what  effect  is  produced 
on  its  reciprocal  ?  its  opposite  ?  the  opposite  of  its  reciprocal  ? 

2.  Which  is  the  larger,  zero  or  a  negative  number  ? 

3.  Explain  the  axioms,  and  illustrate  each  of  them  by  an 
inequality  between  known  numbers. 

Show  which  of  them  are  deducible  from  the  others. 

4.  Prove  the  following  statements,  with  reference  to  larger- 
smaller  inequalities,  giving  axioms  as  authority: 

both  members  of  an  inequality  may  be  multiplied  or  divided 
by  the  same  number,  or  raised  to  the  same  positive  power; 

the  products  of  the  corresponding  members  of  several  in- 
equalities maybe  taken  without  changing  the  sign  of  inequality; 

but  if  the  same  operations  be  performed  on  the  reciprocals 
of  both  members,  the  sign  of  inequality  must  be  reversed. 

5.  Show  by  trial  that  adding  the  same  number  to  both 
members  of  a  larger-smaller  inequality  will  sometimes  reverse 
the  sign  of  inequality. 

So,  subtracting  the  same  number  from  both  members. 
So,  dividing  two  such  inequalities  member  by  member. 

6.  Of  what  numbej-s  are  negative  powers  larger  than  the 
like  positive  powers?  of  what  numbers  are  positive  powders 
smallest  when  the  exponents  are  largest  ? 

7.  Show  that  if  both  terms  of  a  proper  fraction  be  positive 
and  to  both  the  same  positive  number  be  added,  the  fraction 
is  made  larger  thereby. 

So,  that  an  improper  fraction  is  thus  made  smaller. 

Which  number  is  the  smallest: 

8.  ^2,  ^4,  ^8?  9.  >^3,  ^6,  ^9?  10.  |^5,  '^10,  '^^15? 
11.  (1/2)^^  (V4)'/S  (l/8)^/«?  12.  (1/3)'/^  (l/6)'/^  (1/9)^/^? 
13.  (1/5)1/^  (1/10)^/^  (1/1  ^y^''  ?     14.   3 . 5 . 7 . 9,  42 .  8^  6^  ? 


150       INEQUALITIES,   INCOMMENSURABLE  NUMBERS.        [V, 

GREATER-LESS   INEQUALITIES. 

One  number  is  greater  than  another  number  of  the  same 
kind  if  the  remainder  be  positive,  when  from  the  first  the 
other  is  subtracted,  and  less  if  the  remainder  be  negative. 

Of  positive  numbers  the  larger  is  also  the  greater,  but  of 
negative  numbers  the  smaller  is  the  greater;  and  any  positive 
number,  however  small,  is  greater  than  any  negative  number 
of  the  same  kind,  however  large. 

The  signs  are  >,  greater  than  ;  <,  less  than. 
E.g.,    +50<+60,    -30>-80,    +50>-30,    +60>-80. 

AXIOMS. 

8.  If  of  three  numbers  the  first  he  greater  than  the  second, 
and  the  second  he  equal  to  or  greater  than  the  third,  then  is  the 
first  number  greater  than  the  third. 

9.  If  one  number  he  greater  than  another,  and  if  the  same 
number  or  equal  numbers  be  added  to  them,  then  is  the  first 
sum  greater  than  the  other. 

10.  If  07ie  number  be  greater  than  another,  and  if  the  same 
number  or  equal  numbers  be  subtracted  from  them,  then  is  the 
first  remainder  greater  than  the  other. 

11.  If  one  number  be  greater  than  another,  and  if  they  be 
subtracted  from  the  same  number  or  from-  equal  ?iumbers,  then 
is  the  first  remainder  less  than  the  other. 

12.  If  one  set  of  numbers  be  greater  than  another  set  of  as 
many  more,  each  than  each,  then  is  the  sum  of  the  first  set 
greater  than  the  sum  of  the  others. 

13.  If  one  number  be  greater  than  another,  and  if  they  be 
multiplied  or  divided  by  the  same  or  equal  positive  numbers, 
then  is  the  first  product  or  quotient  greater  than  the  other. 

14.  If  one  number  be  greater  than  another,  and  if  they  be 
multiplied  or  divided  by  the  same  or  equal  negative  numbers, 
then  is  tlie  first  product  or  quotient  less  than  the  other. 

NoTK. — The  pnpil  may  compare  these  axioms  witli  those  on 
page  148,  taken  in  order:  he  will  find  that  adding  and  sub- 
tracting in  greater-less  inequalities  are  analogous  to  multiply- 


§3]  INEQUALITIES.  151 

ing  and  dividing  in  larger-smaller  inequalities,  and  that  mul- 
tiplying and  dividing  in  the  one  are  analogous  to  finding 
powers  and  roots  in  the  other.  The  note  at  the  top  of  page 
149  applies  also  to  this  set  of  axioms. 

QUESTIONS. 

1.  Can  a  positive  number  be  less  than  a  negative  number  ? 
can  it  be  smaller  ?  larger  ?  greater  ? 

2.  Name  two  numbers  equally  large,  but  unequal. 

3.  Which  of  the  pair  —(a  +  h),  (a  +  b)  is  the  larger  if  a,  b 
be  both  positive  ?  which  is  the  greater  ?  if  a,  b  be  both  nega- 
tive ?  if  a  be  positive,  b  negative,  and  a  larger  than  b  ? 

4.  In  the  pair  x,  af  which  is  the  larger  if  x>l?  if  a:  be  a 
positive  proper  fraction  ?  if  a:  be  a  negative  fraction  ? 

What  two  values  of  x  make  x^  equal  to  x  ? 

5.  What  axiom  proves  that  in  a  greater-less  inequality  a 
term  may  be  transposed  from  one  member  to  the  other  by 
changing  its  sign  ?  that  if  the  signs  of  every  term  be  changed 
the  sign  of  inequality  must  be  reversed?  that  two  like  in- 
equalities may  be  added  without  changing  the  sign  ? 

6.  Why  do  not  axioms  6, 7  apply  to  inequalities  of  this  kind  ? 

7.  If  a>b  and  a*>V,  the  elements  may  have  such  re- 
lations that  a  —  a'>b  —  b',   or  a  —  a'  —  b  —  b'  or  a  —  a'<b  —  b\ 

8.  If     2a;-7>29,     3.t-5<2:c  +  16,    then     18<a;<21. 

9.  If  16  more  than  three  times  the  number  of  sheep  exceeds 
twice  their  number  and  27,  and  four  thirteenths  of  their  num- 
ber less  one  be  less  than  3,  how  many  sheep  are  there  ? 

10.  Twice  a  number  less  3  is  less  than  the  number  plus  5; 
and  11  plus  twice  the  number  is  less  than  3  times  the  number 
plus  5:  what  is  the  number?  is  the  number  definite? 

11.  \ix  be  any  positive  number,     ^  +  l/a;<2. 

12.  If  x-\-ij-s,  4:Xij-s^-{x-yY,  and  xy  is  greatest 
when  x  —  y  =  0'y  hence  show  that  the  product  of  two  numbers 
whose  sum  is  constant  is  greatest  when  the  numbers  are  equal. 

13.  Show  which  of  axs.  8-14  are  derivable  from  others. 


152       INEQUALITIES,  INCOMMENSURABLE   NUMBERS.  IV 

§  4.    INCOMMENSURABLE   NUMBERS. 

Numbers  that  arise  from  the  effort  to  measure  any  quantity 

by  a  unit  that  has  no  common  measure  with  it  are  i}icommen- 

surable  numbers. 

E.g.,  a  side  and  a  diagonal  of  the  same  square  are  both  definite 
lilies,  but  the  length  of  the  diagonal  cannot  be  ex- 
pressed exactly  by  any  simple  fraction  in  terms  of  the 
side,  nor  that  of  the  side  in  terms  of  the  diagonal; 

i.e.,  neither  line  can  be  got  from  the  other  by  partition  and 
repetition. 

So,  the  circumference  of  a  circle  is  incommensurable  with  its 

diameter. 
So,  a  period  of  time  expressed  exactly  either  in  days,  in  lunar 

months,  or  in  solar  years,  can  be  expressed  exactly  in 

neither  of  the  other  units. 

An  abstract  number  is  incommensurable  if  it  can  be  written 
approximately,  but  not  exactly,  as  a  fraction.  Such  numbers 
arise  from  attempts  to  find  operators  that,  used  in  a  specified 
way,  shall  yield  certain  given  results. 

E.g.,  to  find  an  operator  that,  used  twice  in  succession  as 
multiplier,  will  double  a  unit. 

1.  No  integer  can  he  sucli  operator. 

For  the  operator  =^1  used  twice  as  multiplier  gives  the  unit  as 
result;  ^1  gives  four  times  the  unit;  and  other  integer 
operators  give  still  larger  results. 

2.  No  simple  fraction  can  he  such  operator. 

For,  let  n/d  be  any  simple  fraction  in  its  lowest  terms; 
then  the  concrete  product   uTiit  x  n/d  x  n/d  is   unit  x  ir/d"^, 
and  -.*  the  fraction  n^/d'^  is  irreducible,  [IV,  th.  13  cr.  1. 

.-.  nyd^^%, 

3.  But  a  simple  fraction  can  he  found  that  shall  very  closely 
approach  the  operator  sought. 

For  *.- unit  X  1.4^  =  unit  x  1.9G     and     unit  x  l.S''^ unit  x  2.25, 
.*.  the  operator  sought  lies  between  1.4  and  1.5.       [ax.  2. 


§4]  INCOMMENSURABLE  NUMBERS,  153 

So    •.•  unit  X  1. 41«  =  unit  x  1.9881, 
and      unitxl.422=  unit  x  2.0164, 

.'.  the  operator  sought  lies  between  1.41  and  1.42. 
So  it  lies  between  1.414  and  1.415,  between  1.4142  and  1.4143, 
and  so  on. 
But  such  operator  exists,  and  is  a  definite  number,  which 
is  perfect!}^  expressed  in  the  language  of  geometry  by  saying 
that  it  is  the  ratio  of  the  diagonal  to  the  side  of  the  same 
square,  and  in  the  language  of  algebra  by  |/2. 

QUESTIONS. 

1.  If  two  squares  have  a  side  of  the  same  length,  how  do  the 
two  perimeters  compare  ?  the  two  areas  ?  the  two  diagonals  ? 

If  one  square  be  placed  on  the  other,  how  will  the  diagonals 
lie  ?  If  the  sides  of  the  two  squares  increase  continuously  at 
the  same  rate,  how  do  the  two  diagonals  increase  ? 

2.  If  a  side  of  a  square  be  one  inch,  can  the  lengths  of  the 
diagonals  be  expressed  in  inches  ? 

If  the  side  increase  continuously  to  two  inches,  has  its 
diagonal  been  at  any  time  measurable  in  inches  ?  was  the 
side  measurable  in  inches  at  that  time  ? 

But,  through  all  changes  of  value,  and  from  commensur- 
ability  to  incommensurability,  what  relation  holds  true  be- 
tween the  two  diagonals  ?  between  a  side  and  a  diagonal  ? 

3.  Why  could  not  the  decimal  expression  for  \/2  end  with 
the  figure  1?  2?  3?---9?  with  what  figure  must  it  end  so 
that  the  square  of  it  may  be  2.0  ? 

Show  that  no  decimal  fraction  ends  in  0. 
What  is  thus  proved  as  to  |/2  ? 

4.  Is  the  fraction  2/3  a  commensurable  number?  is  the 
corresponding  decimal  .666  •  •  •  a  commensurable  number? 

Have  these  two  expressions  the  same  or  different  values? 
Is  one  of  them  more  definite  than  the  other? 

5.  Are  the  approximate  decimal  expressions  for  incom- 
mensurable numbers  variables  or  constants  ?  If  variables,  do 
they  grow  continuously  or  discontinuously  ? 


154     INEQUALITIES,    INCOMMENSURABLE   NUMBERS.   [V.Ta 
MULTIPLICATION   AND   DIVISION. 

The  product  of  a  concrete  number  by  an  abstract  number  is 
a  number  that  bears  the  same  relation  to  the  multiplicand 
that  the  multiplier  bears  to  unity. 

This  definition  covers  the  earlier  and  simpler  ones. 
E.g.,  if  the  multiplier  be  2,  the  product  is  the  double  of  the 

multiplicand;  if  ~2,  it  is  the  opposite  of  the  double. 
So,  if  the  multiplier  be  3/4,  the  product  is  the  triple  of  a 
fourth  part  of  the  multiplicand. 
It  also  defines  multiplication  by  an  incommensurable  mul- 
tiplier; and  later  it  defines  multiplication  by  an  imaginary. 
E.g.,  the  product  six  sq.  ft.  x  |/2  is  a  number  that  has  the  same 
relation  to  six  sq.  ft.  that  the  length  of  the  diagonal  of  a 
square  has  to  that  of  one  of  its  sides. 
So,  the  product  6  hours  x  ;r  is  a  period  of  time  that  bears  the 
same  relation  to  six  hours  as  the  length  of  the  circumfer- 
ence of  a  circle  bears  to  that  of  its  diameter. 
So,  if  simple  interest  be  of  continuous  growth,  the  interest 
varies  as  the  time  and    i—p-r-t,    wherein  the  product 
jt?-r  is  the  interest  for  one  year,  and  the  product  p-r-t 
is  the  interest  accrued  at  any  given  time,  t,  whether  t 
stand  for  a  length  of  time  that  is  commensurable  with 
the  unit  year  or  not. 
The  definition  of  the  product  of  two  abstract  numbers  is 
identical  with  that  given  on  page  6;  and  the  definitions  of 
division  and  of  a  quotient  are  identical  with  those  given  on 
page  16. 

MULTIPLICATION   ASSOCIATIVE. 

Theor.  10.  The  product  of  three  or  more  abstract  7iumlers 
(commensurable  or  incommensurable)  is  the  same  fiumber, 
however  the  factors  be  grouped.  [I,  th.  1. 

Let  a,  b,  c  '  '  '  he  three  or  more  al)stnict  numbers; 
then  is  their  product  the  same  number,  however  the  numbers 
be  grouped. 


9,  §4]  INCOMMENSURABLE   NUMBERS.  155 

For,  to  multiply  the  concrete  product  unit  x  a  by  the  ab- 
stract product  bxc  is  to  multiply  the  concrete  prod- 
uct unit  X  a  by  the  abstract  number  b,  and  the  conse- 
quent concrete  product  unit  x  a  x  ^  by  the  abstract 
number  c,  [df.  prod.  ab.  nos. 

and  so  for  other  factors,  and  for  other  groupings  of  them. 

Q.E.D. 
QUESTIOXS. 

1.  Draw  a  square;  then  another  square  whose  side  is  a  diag- 
onal of  the  first  square:  how  does  the  diagonal  of  the  second 
square  compare  with  the  side  of  the  first? 

What  operator,  acting  as  multiplier  on  the  side  of  the  first 
square  and  again  on  that  result,  has  doubled  the  given  side  ? 
Is  that  operator  a  commensurable  number  ? 

Show  that  the  definitions  of  multiplication  and  of  a  multi- 
plier apply  here. 

2.  Of  what  two  numbers  is  the  diagonal  of  a  square  the 
product  ?  Can  the  product,  or  the  quotient,  of  two  incom- 
mensurable numbers  be  a  commensurable  number  ? 

3.  How  long  is  the  diagonal  of  a  square  whose  side  is  one 
inch?  of  a  square  whose  side  is  a  inches?  What  number 
bears  the  same  relation  to  unity  as  the  diagonal  of  a  square 
bears  to  its  side  ? 

4.  What  two  lines  are  proportional  to  7t  and  unity  ? 
Js  TT  a  commensurable  number  ? 

W^hat  perfectly  definite  meaning  has  it  ? 
What  line  is  exactly  expressed  by  47r  ?  10;r  ? 
If  the  circumference  of  a  circle  be  10,  what  is  10/;r?  5/;r  ? 
If  the  radius  of  a  circle  be  4,  what  is  its  area,  the  area  of  a 
circle  being  half  the  product  of  its  radius  and  circumference? 

5.  If  a  side  of  a  square  be  two  feet,  and  if  a  circle  be 
described  upon  its  diagonal  as  diameter,  what  is  the  length  of 
the  circumference  of  the  circle?  what  is  its  area? 

6.  By  what  shall  a  unit  be  multiplied  three  times  in  succes- 
sion to  quadruple  it  ?  four  times,  to  triple  it  ? 


156      INEQUALITIES,   INCOMMENSURABLE  NUMBERS.  [v, 


MULTIPLICATION   COMMUTATIVE. 

Lemma.   Two  rectangles  that  have  the  same  altitude  are  to 
each  other  as  their  bases. 

(a)  The  bases  commensurahle. 
Let  A,  B  be  two  rectangles  of  the  same  height,  whose  bases, 
a,  b,  have  a  common  measure,  c; 


A 


let  c  be  contained  in  a  m  times  and  in  h  n  times;  and  at  the 
points  of  division  draw  perpendiculars  to  a,  b,  thus 
dividing  the  rectangle  A  into  m  parts  and  b  into  n 
parts  that  may  be  shown  to  be  all  equal  by  placing  one 
upon  another; 

then  *.•  A :  B  =  m :  n    and     a:b  —  m\n,  [df.  ratio. 

.'.  K:B  —  a:b.  q.e.d.         [ax.  equal. 

{b)  The  bases  incommensurable. 

Let  A,  B  be  two  rectangles  of  the  same  height,  whose  bases, 
a,  b,  have  no  common  measure; 


A 


then  if  b  be  not  the  fourth  proportional  to  A,  B,  a,  that  pro- 
portional is  some  line  shorter  or  longer  than  b. 

If  possible,  let  that  proportional  be  CD,  shorter  than  b  by  the 
part  DE, 

and  find  a  measure,  c,  of  a,  that  is  shorter  than  de. 

Apply  c  repeatedly  to  b\  then  at  least  one  point  of  division,  F, 
falls  between  D  and  e; 

draw  perpendiculars  to  a,  b  as  in  («); 

then  •.•  «,  CF  are  commensurable,  [constr. 

.*.  A :  rectangle  on  cf  =  « :  CF.  ,  (a) 


But       A:B  =  a:CD; 


\}^yV' 


Th.  10§4]  INCOMMENSURABLE  NUMBERS.  157 

and  •.*  a: rectangle  on  cf>a:  B     and     «:CF<rt:CD,      [ax.  4. 

.-.  these  two  proportions  are  not  both  true, 
i.e.,  the  hypothesis  that     A:B  =  rt:some  line  shorter  than  h 

leads  to  a  false  conclusion^  and  is  itself  false. 
So,  as  nijiy  be  proved  in  like  manner,  the  hj^pothesis  that 
a:  B=:rt  :  some  line  longer  than  b  is  false, 
.*.  it  is  only  left  that  A\B  =  a:b.  q.e.d. 

QUESTIOXS. 

1.  What  is  a  lemma?  [consult  a  dictionary. 

2.  In  case  (a),  let  c  be  one  of  the  small  rectangles  into  which 
A,  B  are  divided ;  express  a,  b  in  terms  of  c>  and  show  why 
A:B  =  m:?2,     and  why     a\})  —  m\n. 

3.  If  a  be  an  incommensurable  line,  can  c  be  an  exact 
measure  of  a  ?  and  if  c  be  a  measure  of  a,  how  can  a  be  called 
incommensurable  ? 

4.  In  case  {h)  why  must  F  fall  between  D  and  e  ?  Give  a 
reason  for  the  two  inequalities  stated.  How  do  these  in- 
equalities prove  that  both  proportions  can  not  be  true  ? 

How  is  it  known  which  is  the  false  proportion? 

5.  Such  a  proof  as  that  of  this  lemma  is  called  an  indirect 
proof,  because,  instead  of  proving  directly  what  we  wish  to 
establish,  we  prove  that  every  other  possible  supposition  leads 
to  a  false  conclusion,  and  is  therefore  itself  false :  why  is  this 
proof  also  called  a  proof  by  exclusion  f 

6.  Construct  two  rectangles  whose  areas  are  proportional  to 
a  side  and  a  diagonal  of  a  square;  two,  whose  areas  are  pro- 
portional to  the  diameter  and  circumference  of  a  circle. 

7.  By  aid  of  the  well-known  theorem  of  geometry,  "the 
square  of  the  hypothenuse  of  a  right  triangle  is  the  sum  of 
the  squares  of  its  legs,"  construct  a  line  whose  length  is 

^3,     |/5,     4/6,     V^,     V^,     i/10,     yll,     V12,     V13. 

8.  Construct  a  square;  a  square  on  the  diagonal  of  tins 
square;  a  third  square  on  the  diagonal  of  the  second  square; 
and  a  fourth  square  on  the  diagonal  of  the  third:  what  rela° 
tion  have  the  sides  of  these  squares?  what,  their  areas? 


158     INEQUALITIES,    INCOMMENSURABLE   NUMBERS.   LV,Tn, 

Theor.  11.  Tlie  product  of  two  or  more  abstract  numbers 
(commensurable  or  incommensurable)  is  the  same  number,  m 
whatever  order  the  factors  be  multiplied. 

(a)  two  factors  a,  b. 
For,  let  u  be  a  square  whose  side  is  of  unit  length,  p  a  rectan- 
gle of  height  1  and  breadth  «,  Q  a  rectangle  of  breadth 
1  and  height  5,  r  a  rectangle  of  breadth  a  and  height  b; 


^  1, 

1 

a 

Q    b 

R 

1  a 

then'/p  =  ux«,     q  =  jjxb,     R  =  Px5  =  Qxa,  [lem. 

wherein  a  is  the  ratio  of  line  a   to  the  unit  line, 
and  so  of  b, 
.*,  R  =  U  X  rt  X  Z>  =  u  X  5  X  «, 

,*.  the  abstract  products    axb,    bxa,   do  the  same  work 
and  are  equal. 
(b)  three  or  more  factors  a,  b,  c. 
The  proof  is  identical  with  that  of  I  theor.  2  {d). 

AXIOMS   AND   THEOREMS   THAT   APPLY  TO   INCOMMENSURABLE 

NUMBERS. 

The  quality  of  numbers  as  denoted  by  the  positive  and 
negative  signs  applies  alike  to  commensurable  numbers  and 
to  incommensurable  numbers. 

The  axioms  of  equality  and  of  inequality,  and  the  definitions 
and  principles  of  division  and  reciprocals,  are  the  same  for  in- 
commensurable numbers  as  for  commensurable  numbers;  and 
I  theors.  3,  4,  relating  to  reciprocals  and  division,  apply  to 
incommensurable  numbers  without  change  in  their  statement 
or  proof. 

So,  theor.  5,  that  addition  is  commutative  and  associative, 
has  the  same  statement,  and  the  proof  is  as  follows: 
For,  let  a,  b,  c  ' '  '  ha  any  abstract  numbers,  commensurable 

or  incommensurable, 
let  these  numbers  act  as  operators  on  any  unit, 
and  let  the  results  be  grouped  and  added  in  any  wayj 


10,  §4]  INCOMMENSURABLE   NUMBERS.  159 

then  *.•  the  quantities  so  found  are  of  the  same  kind, 
and        their  aggregate  is  the  same,  in  whatever  order  they  be 
arranged  and  however  they  be  grouped, 
,*.  the  several  sums  of  these  operators  do  the  same  work 
and  are  all  equal.  q.e.d. 

So,  theor.  7,  that  multiplication  is  distributive  as  to  addi- 
tion; theors.  8,  9,  relating  to  opposites  and  subtraction;  and 
theors.  10,  11,  12,  relating  to  integer  powers,  apply  to  incom- 
mensurable numbers  without  change. 

QUESTIONS. 

1.  Show  that  the  proof  of  theor.  11  could  not  have  been 
complete  without  the  aid  of  the  preceding  lemma. 

2.  State  the  proof  of  theor.  11  for  three  or  more  factors. 

3.  What  modification  is  made  in  I  theor.  5,  to  make  it  ap- 
plicable to  incommensurable  numbers  ? 

4.  Keview  all  the  theorems  that  apply  without  change  to 
incommensurable  numbers  and  give  the  proofs,  remembering 
that  they  include  such  numbers. 


160        INEQUALITIES,  INCOMMENSURABLE  NUMBERS.        [V, 

§  5.  QUESTIONS   FOR  REVIEW. 

Define  and  illustrate: 

1.  A  constant;  a  variable;  related  variables;  a  function  of 
a  variable;  continuous  variables;  discontinuous  variables. 

2.  Variation;    direct   variation;    inverse   variation;    joint 
variation. 

3.  The  ratio  of  one  quantity  to  another  quantity  of  the 
same  kind ;  the  ratio  of  two  abstract  numbers. 

4.  A  direct  ratio;  an  inverse  ratio. 

5.  Proportion;  proportionals;  a  continued  proportion. 

6.  An  antecedent;  a  consequent;  the  extremes;  the  means. 

7.  A  mean  proportional;  a  third  proportional. 

8.  An  inequality;    larger-smaller  inequalities;   greater-less 
inequalities. 

9.  A  product  where  the  multiplier  is  an  incommensurable 
number;  a  quotient  where  the  elements  are  incommensurable, 

10.  A  lemma;  an  indireot  proof. 

State,  with  any  neccss.iry  explanations  or  illustrations: 

11.  Tiie  axicms  of  larger-smaller  inequality. 

12.  The  axioms  of  greater-less  inequality. 

13.  How  to  find  a  missing  proportional. 

14.  The  theorems  that  apply  to  proportions  involving  con- 
crete numbers. 

Prove  that : 

15.  The  product  of  the  extremes  of  a  proportion  equals 
that  of  the  means. 

IG.  The  factors  of  two  equal  products  form  a  proportion. 

17.  A  proportion  may  be  written  by  alternation;  by  inver- 
sion; by  composition  ;  and  by  division. 

18.  The  terms  of  a  proportion  may  be  multiplied  or  divided 
by  the  like  terms  of  another  proportion. 

19.  The  terms  of  a  proportion  may  be  multiplied  or  divided 
by  the  same  number. 


$51  QUESTIONS  FOR  REVIEW.  161 

20.  The  antecedents  of  a  proportion  may  be  multiplied  or 
divided  by  the  same  number,  and  so  may  the  consequents. 

21.  The  first  and  second  terms  of  a  proportion  may  be 
multiplied  or  divided,  by  the  same  number,  and  so  may  the 
third  and  fourth  terms. 

22.  The  terms  of  a  proportion  may  be  raised  to  the  same 
power,  and  the  same  root  may  be  taken  of  them. 

23.  In  a  continued  proportion  the  sum  of  the  antecedents 
is  to  the  sum  of  the  consequents  as  any  antecedent  is  to  its 
consequent. 

21.  Two  rectangles  of  equal  altitude  are  as  their  bases. 

25.  Variation  by  a  fixed  law  is  expressed  by  an  equation. 

26.  A  proportion  is  an  expression  of  direct  variation. 

27.  Addition  is  a  commutative  and  an  associative  operation 
with  incommensurable,  as  with  commensurable,  numbers. 

28.  What  operations  may,  and  what  may  not,  be  performed 
on  the  two  members  of  a  larger-smaller  inequality?  of  a 
greater-less  inequality? 

29.  State  and  prove  the  associative  and  commutative  prin- 
ciples of  multiplication,  with  incommensurable  numbers. 

30.  The  sum  of  the  gi-eatest  and  least  of  four  positive  pro- 
portionals is  greater  than  the  sum  of  the  other  two. 

What  is  the  law  if  some  of  the  proportionals  be  negative  ? 

31.  If  X,  y  be  such  numbers  that  a  ^-  x  \  h  -\-  y  =  a  \  b, 
then     X  :  y  —  a  \  h. 

32.  The  sum  of  a  real  positive  number  and  its  reciprocal 
cannot  be  less  than  2. 

33.  If  X  '.  y  =  {x  -\-  zf  :  {y  +  zf,  2;  is  a  mean  proportional 
between  x  and  y. 

34.  If  a  :l)  =  c  :  d  and  Z*  be  a  mean  proportional  between  c 
and  (/,  then  c  is  a  mean  proportional  between  a  and  h. 

35.  If  rtaZ/",     ^  a  c^,     c  a  d%     then  a  a  d^°. 

36.  lixccy- ",  and  x  =  1/2  when  ?/  =  4,  what  is  x  when  y  =  2/3? 

37.  lia:b  =  h',c^c:d,  then     a\d^a'\l\ 


162  POWERS   AND   ROOTS.  tVI,TH. 

VI.  POWERS  AND  ROOTS. 


The  words  poioer,  root,  base,  exponent,  root-index  are  defined 
in  I,  §  5.  A  root-index  is  always  assumed  to  be  a  positive 
integer;  but  an  exponent  may  be  any  number  whatever. 

§  1.    THE   BINOMIAL  THEOREM. 

Theoe.  1.  If  a  Unomial  he  raised  to  any  positive  integer 
poiver,  that  poiuer  is  symmetric  as  to  the  tivo  terms  of  the 
binomial,  and  consists  of  a  series  the  number  of  luhose  terms 
is  one  greater  than  the  exponent  of  the  binomial;  and  the  suc- 
cessive terms  of  this  series  are  the  products  of  three  factors : 

1.  Coefficioits  that  come  from  the  exponent  of  the  binomial: 
the  first,  one;  the  second,  the  exponent;  the  third,  the  product 
of  the  second  by  half  the-exptonent-less-one;  the  fourth,  thej^rod- 
uct  of  the  third  by  a  third  of  the-exponent-less-tioo;  and  so  on, 

2.  Falling  poivers  of  the  first  term,  beginning  with  that 
power  whose  exponent  is  the  exponent  of  the  binomial. 

3.  Rising  poiuei^s  of  the  secoiid  term,  beginning  with  the 
zero-poioer. 

Let  a,  b  be  any  two  numbers,  then  the  theorem  is  written 
{a  +  bY  =  a''-\-na''-'^b  +  ln{n-\)a''-''b^ 

+  \n{n-l){n-'Z)a''-^b^+  •  •  •  +5% 
and  it  is  proved  by  inductio7i,  as  shown  below. 

(a)   The  laiu  is  true  for  the  second  power. 
For  •.•(«  +  bf  =a^-\-  lab  +  b^,  [mul tiplication. 

.-.  {a-\-bY-a''^-na''-^b^-  •  •  •  +J",  when    ?i  =  2. 

(^)  If  the  law  be  true  for  any  one  power  it  is  true  for  the 
next  higher  power. 

For,  assume     {a^-bf  =  a^  +  ha^-''b  +  ^h{k-l)a^-^b-\-'"+h^, 
and  multiply  both  members  of  this  equation  by    a  +  b, 
then  (a-^bf^^  =  a'''-^  +  {h  +  l)aH  +  ^{h-\-l)Jca^-'b''+  -  •  •  +^^+» 
a  series  of  the  same  form  as  tliat  for  tlie  A'th  power,  ^'  +  1  tak- 
ing the  place  of  k,  and  conforming  to  the  law  of  development. 


1,§1]  THE  BINOMIAL   THEOREM.  163 

(c)   TJie  latv  is  true,  whatever  the  ])ower. 

For  •.•  it  is  true  for  the  second  power,  [(«) 

.-.  it  is  true  for  the  third  power;  \{h) 

and  •.•  it  is  true  for  tlie  third  power,  [above 

.-.  it  is  true  for  the  fourth  power,  and  so  on.  q.e.d. 

QUESTIONS. 

1.  In  any  power  of     {u  —  h)     what  terms  are  negative  ? 
In  what  powers  is  the  last  term  negative  ?  positive  ? 
Expand : 

2.  {x  +  yf,         3.  {a-^y.         4.  {a  +  h)\         5.  {2x-y)\ 
6.  {x^+^y)K      7.  (a-'^bcy,     8.  {a±l)f,        9.  (m-n)\ 

10.  {^/{m  +  n)]K      11.  [-'^/(x-y)Y.      12.   {^x-\-%y)K 
13.  (2a;/3-3/2af.  14.  {^x  +  ^af,  15.   {4.ax  +  dy-l)\ 

16.  {Zc-^Uy.  17.  {x  +  y-^zf.  18.   {:x-y  +  4.zf, 

19.  [{^a-^  +  ic-d)^.  20.   l{^x  +  y)-{:x-2y)Y. 

21.  A  proof  by  induction  consists  of  three  steps:  name  them. 

22.  In  the  expansion  of    {a-]-hY^,    and  of    (a  —  bf^,    write 
the  literal  part  of  the  third  term;  of  the  50th  term ;  of  the  51st. 

23.  What  term  of  {a^hf  \s> 

n{n-l)"-(n-r  +  \)'ce'-'''h''/r\'^     [r!  =1.2.3.. -r. 
How  may  the  last  factor  in  the  numerator  of  a  coefficient 
be  found  from  the  number  of  the  term?  in  the  denominator? 

24.  Of  what  term  of    {a  +  hf^    is    50-49.48.  . -43/8!    the 
coefficient  ?     50 • 49 • 48 • • . 9/42 !  ? 

25.  What  other  term  has  the  same  coefficient  as  the  49th  ? 
the  50th  ?  the  41st  ?     What  is  the  general  principle  ? 

Make  use  of  the  general  formula  in  ex.  23  to  write: 

26.  The  fourth  term  of     {x-bf'-,  the  7th  term;  the  12th. 

27.  The  12th  term  of     {l-^x'y^',  the  third  term;  the  8th. 

28.  The  middle  term  of     {a/x  +  x/ay''',  the  third;  the  7th. 

29.  In     (x-^-yY^     the  sum  of  the  coefficients  of  the  odd 
terms  equals  the  sum  of  tlic  coefficients  of  tlie  even  terms. 

30.  AVhat  term  of     {x  +  l/xf^'    is  free  from  x  ? 


164  POWERS  AND  ROOTS.  [VL 

§2.   FRACTION  POWERS. 

K  fraction  power-  of  a  number  is  either  a  root  of  the  num- 
ber or  some  integer  power  of  a  root.     The  record  of  the  oper- 
ation begins  by  naming  the  base,  then  the  number  of  factors 
it  is  resolved  into,  then  the  number  of  such  factors  that  are 
multiplied  together.     The  two  numbers  last  named  appear  as 
tho  denominator  and  the  numerator  of  a  simple  fraction. 
E.g.,  642/^=(  V'64)2  =  4-4=:16,     wherein   64   is   resolved  into 
the  three  equal  factors  4,  4,  4,  and  the  product  of  two 
of  them  is  taken. 
So,  64-2/^  =  (|/64) -2=  1/42  =  1/16,      wherein   64    is   resolved 
into  the  three  equal  factors  4,  4,  4,  and  two  of  these 
factors  are  used  in  partition. 
Note  the  new  use  of  the  fraction  form  2/3 :  As  an  exponent, 
it  means  that  64  is  resolved  into  three  eqiml  factors,  and  two 
of  them  are  multi2:)lied  together;  in  the  ordinary  usage,  some 
unit  is  divided  into  three  equal  parts,  and  t2vo  of  them  are 
added  together.    Later  it  appears  that  these  fraction  exponents 
are  subject  to  the  laws  already  established  for  fractions;  but 
this  must  not  be  assumed  without  proof. 

Integer  powers  and  fraction  powers  are  classed  together  as 
commenstiraUe  potoers;  incommensui-able  powers  appear  later. 

The  value  of  a  fraction  power  is  often  ambiguous. 
E.g.,  100i/^=±10;    9-^/2=  ±1/2";    (a^Y^'=±a;    (a'Y/'=  ±a\ 
Different  powers  of  a  base  are  in  the  same  series  if  they  be 
integer  powers  of  the  same  root.     An  integer  power  of  a  base 
belongs  to  all  series  alike. 

91,    9^/^   9^  ...  are  the 
2d,   3d,    4th,  • . .  powers  of  |/9, 
9,    -27,    81,...,  powers  of -3 
9,     27,    81,..-,  powers  of +3 
When  several  powers  of  the  same  base  occur  together,  they 
are  assumed  to  be  all  taken  in  the  same  series. 

Powers  that  have  the  same  exponent  are  like  powers. 
E.g.,  |/«,  yb,  ^ah\    a\  h\  ^';    2»,  3",  G";   o;"^/*,  y-^/*,  z-'^^K 


E.g.,    9-', 

9 -■''»,    9»,      9'/^ 

-2d,   ■ 

-1st,      0th,  1st, 

i.e.,      1/9, 

-1/3,     1,    -3, 

and      1/9, 

1/3,      1,       3, 

»21  FRACTION   POWERS.  165 

QUESTIONS. 
Explain  the  meaning  of  each  part  of  the  expressions  below. 

1.  -27^/3.  2.  -27-2/^  3.  -27*^.  4.  Sp/*. 

5,  81-^/*.  6.  243^/5.  7.  -2^3'^.         8.  243^/5. 

9.  -243^/^      10.  243-*/^.  11.  Sl-^^l  12.  1728^/3^ 

13.  What  operation  is  indicated  by  the  denominator  of  a 
simple  fraction?  what  by  the  numerator?  what  by  the  terms 
of  a  fraxition  exponent  ?     Can  it  be  assumed  that  fractions,  as 
exponents,  are  operated  with  just  as  other  fractions  ? 
Express  with  fraction  exponents : 

14.  |/.r5.  15.  ^a\  16.  i/a'l^c. 
17.  ( ^fy.             18.  4  ^/ah<?/x,         19.  ^{xyz/W). 

20,    (3f626u)\         2L   ij%a  ,^{x/y\  22.  v'(:c  +  2/)/(«-2^)'- 
Express  with  radical  signs: 

23.   (a^/2f.  24.  {h-^'f\  25.  (c-^)^/". 

26    (.r3//2)-2/3.       27.  (5«a;-2/3)-2''.  28.  3/a^if^'/4:?n^/K 

29.  a2/3/j-'/*.  30.  8«-i/3_^j-2/r^  3^^  a;^/^  +  3/^/7^/*. 

Express  with  positive  exponents : 

32.  a;-3.  33.  6cr''x-\  34.  a-^-''^, 

41.  Is  the  square  root  of  an  incommensurable  number  a 
commensurable  or  an  incommensurable  power?  what  is  2<^? 

42.  What  kind  of  powers  give  rise  to  two  values?  what 
powers  of  these  powers  are  alike  in  botli  series  ? 

Find  the  first  five  powers  of  16^^^  in  both  series. 
Find  the  value,  or  values,  of: 

43.  27-^/^ ~27i/'+( -27)2/1  44.  25^/2_^25-3/8  +  25«. 

45.  163/*  +  16^/*-16-^/*-16-5/*.  46.  8-^/^8-2-8-/^8-^ 
47.  32^/^-322/^32-^/^32 -*'^  48.  36i/2_  3(33/2^03-5/3^ 


166  POWERS  AND  ROOTS.  IIV.  Tna 

A  COMMENSURABLE   POWER  OF  A  COMMENSURABLE   POWER. 

Theor.  2.  A  commensurable  power  of  a  commensurable 
poiuer  of  a  base  is  that  poiuer  of  the  base  whose  exponent  is  the 
product  of  the  ttoo  given  exponents. 

Let  m,  n  be  any  two  commensurable  numbers,  and  A  any  base; 
then  (a"^)"  =  a"^". 

{a)  m,  n  integers,  positive  or  negative,  [I,  th.  11. 

(&)  m,  n  reciprocals  of  positive  integers,  \/p,  1/q, 

For,  resolve  A  into  p  equal  factors  B,  b,  •  •  • ,  commensurable  or 

incommensurable,  so  that  B  =  A*/^, 
and  resolve  B  into  q  equal  factors  c, c,  •  •  •,  so  that  c=  (a*/**)*/'; 
then  •/  A  is  thus  resolved  into  pq  equal  factors  c,  c,  •  •  •  , 

/.  (a^/^)*/«=a*/^.  q.e.d.         [df.  fract.  pwr. 

(c)  m,  n,  simple  fractions,  p/q,  r/s,  and  q,  s  positive. 
For,  resolve  A  into  q  equal  factors  B,  B,»  •  -and  take  p  of  them, 
and  resolve  B  into  s  equal  factors  c,  c,*  •  -and  take  r  of  them; 
then  •/  A  is  thus  resolved  into  qs  equal  factors  c,  0,  •  •  •  and  pr 
of  them  are  taken, 
.*.  (a^/«)'"/*=a'^/«».  q.e.d.         [df.  fract.  pwr. 

And  so  if  three  or  more  powers  be  taken  in  succession. 
CoR.1.  (a"^)'/"  =  a"*/". 
Cor.  2.  (a")^"  =(a»^»)«=a. 

equal  fraction  powers. 

Theor.  3.  TVie  value  of  a  fraction  potoer  of  a  base  is  the 

same,  lohether  the  fract  ion  exponent  be  in  its  lowest  tenns  or  not. 

For,  let  h,  p,  q  be  a!iy  positive  integers,  A  any  base, 

then    A^/^'^=\[(A}^'^Y/^f\^  [th.  2. 

=  (a»/«)''  [th.  2  cr.  2 

=  A<'/'.  Q.E.D.         [df.  fract.  pwr. 


•^,3,  §2]  FRACTION   POWERS.  167 

QUESTIONS. 

1.  Explain  the  proof  of  theor.  2,  and  tell  liow  it  applies 
when  A  is  not  a  perfect  power  of  the  pqth  degree. 

Find  the  values  of: 

2.  (83/«a-3)2/3.  3.  4(a;-2/3)3/2.  4.  (642/^)-V6. 

5.  i/(a^bc  ^a^c)\  6.  (9^^^)^  7.  (125-2/3) -3/2. 

8.  [(Aa^-12x  +  9y/^f.  9.  (16a-' /81b') -^^K 

10.  (92^/25?/2)-n  11.  (256/625) -3/*. 

12.  p{a^b/paby,  13.  f^-^^/rc-i/y/^ 

14.  Resolve  64  into  six  equal  factors  and  indicate  the  con- 
tinued product  of  twenty-five  of  them;  then  resolve  64  into 
two  equal  factors,  resolve  the  product  of  five  of  these  factors 
into  three  equal  factors,  and  indicate  the  prime  factors  in  the 
product  of  five  of  them:  hence  show  that     (64^/^)^/3=: 6 4^^/^ 

15.  In  the  expression  a^'''^,  can  the  exponent  kp/kq  be 
replaced  by  the  equal  fraction  p/g  without  further  proof? 

16.  Resolve  729  into  three  equal  factors  and  take  the  product 
of  two  of  them;  then  resolve  729  into  six  equal  factors  and  take 
the  product  of  four  of  them:  hence  show  that  729^^3 _»;'29*/^ 
Simplify  the  radicals: 

17.  a^^\  18.  (a  +  b)-^^,       19.  (x-yY^^^K     20.  a^^-^K 
21.  ^a\          22.  ^{a  -  by.         23.  ffq-\         24.  '^{x'/f). 
25.  {x^i/y.   26.(x-'/^/y-^/%  27.  (rt^/yl/a^f.   28.  (8f^ -3) -2/6. 

29.  How  can  fraction  powers  of  different  degrees  be  reduced 
to  powers  of  like  roots  ?  are  their  values  changed  thereby  ? 
Reduce  to  powers  of  the  twelfth  root : 

30.  x'/\  Sx"^',  a>'\  31.  (a;  +  ?/)2/3,  yV\  32.  -J-i/',  c^,  a^'\ 
33.  a;-^  1/'/^,  2-^/^  34.  {x^yy^,  y-^'K  35.  h"^^,  c-^,  a-'^'K 
36.  ^/x,  y/ij,  ^^.  37.  (  |/x- 1/?/)2/3,  2-2/3.   38  («i/3)-i/4^  («-V3)i/*^ 

Reduce  to  powers  of  like  roots: 

39.  "yx,  Yy.  40.   "^^a^l^cr,  ""^a,  41.  ^2,  |/3. 

42.   i/ab,  ;/c.  43.  x'/%  y^^K  44.  a;- ^/2,  y- 2/3, 


168  POWERS   AND   ROOTS,  [VI,Ths. 

PRODUCT   OF   COMMEiTSURABLE   POWERS   OF  THE   SAME   BASE. 

Theor.  4.  The  product  of  two  or  more  commensurahle poivers 
of  a  base  is  that  power  of  the  base  whose  exponent  is  the  stwi 
of  the  expo7ients  of  the  factors. 

Let  m,  n  be  two  commensurable  numbers,  and  A  any  base ; 
then     A"^-A"  =  A'"+", 

{a)  m,  n  integers,  positive  or  negative,  [I,  th.  10. 

{b)  My  n  simple  fractions,  p/q,  r/s,  and  q,  s  positive. 
For      A^/« '  A^/* = A^"/«*  •  A«''/«"  [th.  3. 

—  j^ps/qs+qr/qs  F^J^^  2, 

-jj>/q  +  r/s^  Q^E.D.  [th.  3. 

And  SO  for  three  or  more  powers. 
CoR,  aVa'^^^a'^-^ 

PRODUCT   OF    LIKE    COMMENSURABLE    POWERS   OF    DIFFERENT 

BASES. 

Theor.  5.  The  product  of  like  commensurable  powers  of  two 
or  7nore  bases  is  the  like  poiver  of  their  product. 
Let  n  be  any  commensurable  number,  and  A,  E,  c,  •  •  •  any 

bases ; 
then    A^'-B'^-c"  .  ••  =  (a-b.c  • .  •)". 

(a)  n  an  integer,  positive  or  negative,  [I,  th.  12. 

{p)  n  a  simple  fraction,  p/q, 
¥ov\'AP'B'p={K'BY,  [{a), 

and     aF={aP^^Y,     b^=(b^/«)«,  [th.  2. 

.•.(ap/«)'^.(b^/'^)''.-=(a.b)^ 
and  •.•  {k^/^Y-  (b^^/^^)'^  =  (ap/<'  •  B^/^)'^,  [(a). 

.•.(ap/«.b^/'^)'^=(a-b)^; 

.-.  A^/« .  B^/'^  =  (a  .  B)^/*^.  [in  ax.  7. 

Cor.  AVB"=(A/B)^ 


4, 5,§a]  FRACTION  POWERS.  169 

The  qth  roots  involved  in  the  last  statement  must  be  used 
with  some  caution  as  to  the  signs:  the  equation  means  that 
the  real  positive  roots,  if  any,  are  equal,  and  that  all  the  roots 
have  the  same  magnitude. 

QUESTIONS. 

1.  Does  the  proof  of  theor.  4  differ  from  that  given,  when  A 
is  an  incommensurable  number? 

2.  Prove  theor.  4  for  the  product  of  three  or  more  powers. 

3.  Assume    a*"/a"^a"*-",    and  multiply  both  members  by 
A":  if  the  result  be  contradictory  to  theor.  4,  what  is  proved  ? 

What  is  such  a  method  of  proof  called? 
Write  the  expressions  below  in  their  simplest  forms,  using  only 
positive  exponents: 

4.  8p/*-81-i/l       5.  Sl^/ySl-*/*.       6.  2565/«.256-3/V256i/l 
7.  ah'^^ca-y^hc.    8.  a^^h/a-'^Hc.    9.  x'''^y"'z^''''X-^"^y-^'K 

10.  [{(a-^)-"}^]«:[!(0"}"T~*-     11-  a;^/y/V/y:2;-3/«y-i/*. 

12  f^^V''.f^V''.r^V''         13  0^-^)"'-(^+.yF 

14.  Prove  that    A  "-p/^  •  B  -^/«  •  c  -^"^  =  ( A  •  B  •  c)  -^/«. 

15.  How  many  signs  has  A^/*,  or  B^^^,  when  q  is  odd?  even  ? 
In  each  case  what  is  the  sign  of  the  product    a^/^-b^^^  ? 
Show  that  in  any  case,  a  value  of  (a-b)^/^  has  the  same  sign. 

16.  Multiply  the  product  of  two  of  the  three  equal  factors 
of  8  by  the  product  of  two  of  the  three  equal  factors  of  125, 
and  compare  the  result  with  the  product  of  two  of  the  three 
equal  factors  of  the  product  8-125:  write  the  conclusion, 
using  fraction  exponents. 

Find  the  values  of: 

17.  81*/*. 256^/*.  18.   642/3.1252/3.         19.  64-2/yl25-2/^ 
20.  (r^.^f/"^-l/^           21.  t^x?'  ^if.  22.    V^V  V^/- 

23    [  ^~^  yV^^'-^'V  94    rr^l-ny  fn-mV 

'  W  +  ab)    W  —  abJ'  '^  '  \  x  —  y  I  '\x  —  yj  ' 

25.  If  a^  =  ^%  then  (a/^>)«/^  =  a«/^-i;  and  if  «  =  2 J,  then  b  =  2. 


170  POWERS   AND   ROOTS.  [VL 

§3.    RADICALS. 

A  radical  is  an  indicated  root  of  a  number.  There  may  be 
a  coefficient;  and  then  the  whole  expression  is  called  a  radical, 
and  the  indicated  root  is  the  radical  factor. 

Any  expression  that  contains  a  radical  is  a  radical  exjjression. 

A  radical  is  rational  if  the  root  can  be  found  and  exactly 
expressed  in  commensurable  numbers;  irrational,  if  the  root 
cannot  be  so  found.  It  is  real  if  it  do  not  involve  the  even 
root  of  a  negative;  imaginary,  if  it  involve  such  root. 

An  expression  that  contains  an  irrational  radical  is  a  surd. 

E.g.,    f256,     \/S,    1^-8,     ^a\    \/{a''-^2ab  +  W)    are  radicals 
with  the  rational  values    ^2,   2,    ~2,    a,    "^(a  +  b)-, 

but      \/x,    y/a^,  \/a\    ^a-a'^^^,    ^a^  +  b^f^^    are  irrational; 
and,  while  all  these  radicals  are  real, 

|/-1,    \/~a^,    \/~2a^,   (~aY^^,  a  +  b  \/~l   are  imaginary. 

Roots  of  rational  bases,  and  integer  powers  of  such  roots, 
with  rational  coefficients,  if  any,  are  simple  radicals. 

The  degree  of  a  simple  radical  is  shown  by  its  root  index. 

A  simple  radical  is  quadratic,  cubic,  quartic,  (biquadratic) 
• .  •  when  the  root  index  is  2,  3,  4,  •  •  • 

E.g.,    l(a^  +  b^y/\  U¥'  ^{a'-bc'),  a^-a^^^  are  simple  quad, 
ratio,  cubic,  and  quartic  surds  in  their  simplest  forms; 

but  *  y^^,   y/a^,  >^/8,   4/" 8,  |(«V  +  Z'V)^/'',  are  simple  radicals 
not  in  their  simplest  forms;  they  may  be  written: 

^a^a,      a^a,      =^24/2.      =^2|/-2.     ^ic(a'  +  bY'. 
Two  radicals  are  like  (similar)  if  they  have  the  same  radical 
factor;  they  are  conformable  if  they  can  be  made  like;  7ion- 
confor7nable  if  they  cannot  be  made  like. 

E.g.,    2x{a''-\r¥Y/\     8(a;-^)(^/HZ'')'/'    are  like  radicals, 
and      |/18,      |/32,      4/98     are  conformable. 

The  sum  of  two  non-conformable  simple  surds,  or  of  a 
rational  expression  and  a  simple  surd,  is  a  binomial  surd;  the 
sum  of  three  non-conformable  simple  surds,  or  of  two  such 
surds  and  a  rational  expression,  is  a  trinomial  surd* 


§3J  RADICALS.  171 

QUESTIONS. 

Are  these  radicals  rational  or  irrational?  real  or  imaginary? 

1.   ^D.  2.   |/343.  3.  |/-256.  4.  |^-243. 

5.   ^{x^-f).   6.  \/(l-2  ^x-\-x).  7.  ^2564/-!.      8.^^-216. 

9.  What  power  of  a  radical  is  sure  to  be  rational  ? 
Tell  why  the  statements  below  are  true : 
10.  x'^'  =  x'''x'^K  11.    p-4:=-  ^4:.       12.  4/-4^-  |/4. 

13.  4|/3=  |/48.         14.  3  4/-4=  ^-36.    15.    -34/4=  4/36. 

16.  |/(2/5)=  V(10/25)=  V(l/25).  |/10=±^V10. 

17.  -'dx^bct'b^  |/-27:c^.  -^5«2^=  v^-135ff2^a;3. 

18.  4/420«^=±2rt.  |/105«.  19.    ^(5/48)  = -j-i^  ^180. 
Keduce  the  radicals  below  to  their  simplest  forms. 

20.  \/2^^.  21.  V-169.        22.   ^/lQ(a  +  h).        23.   -^729. 

24.  81 -^         25.  495/1  26.  500^/^^  27.  ^00^^^, 

28.  Change   the   quadratic  surds   on  the   opposite   page   to 

quartic  surds,  and  the  cubic  surds  to  sextic  surds. 

29.  |/20,    |/45,     4/(4/5)     are  conformable  surds. 

30.  Change    4/2,     \/^     to  surds  of  the  same  degree. 

If  possible  make-  like  the  radicals  below,  and  reduce  the  ex- 
pressions to  their  simplest  form : 

31.  ^^^32  +  6^/2.    32.   4/3+  4/1       33.  x'^^-\-x'/\   34.  j^x-  ^y. 
35.  5  |/98.^  + 104/22;.  36.  (3«''Z>)i/2-(27a2Z>y/2. 

37.  V(5a;-5)-  4/(2:?; -2).       38.   ^'(8^^^  + 16«*)  -  x^{b'^  +  'ilaW), 
39.  (36f^^?/)i/^-(25?/)n  40.  2/a;-3/2- J«/ 4/^?;. 

41.  Separate  592704  into  its  prime  factors:  which  of  them 

occur  three  times  ?  what  is  the  value  of    -^592704  ? 

42.  So,      ^78400;      y'50625;     -^^27225;     >^3111696. 

43.  Are     \/{a  —  h)     and     y^    rational  or  irrational  ? 
Replace  a  by  22,  ^  by  6,  ?/  by  \/2 :  what  changes  are  made  ? 
Can  a  literal  expression  be  imaginary  and  its  numerical  value 

be  real,  or  the  reverse  ? 


172  POWERS   AND   ROOTS.  [VI,Prs 

operatio:n^s  oi^  radicals. 

PrOB.  1.    To   REDUCE   A   RADICAL  TO   ITS   SIMPLEST   FORMo 

Resolve  the  number  ivhose  root  is  sought  into  two  factors: 
one  the  highest  possible  perfect  poicer  of  the  same  degree 
as  the  radical,  and  the  other  an  e^ilire  number; 

write  the  root  of  the  first-named  factor  as  a  coefficient  before 
the  indicated  root  of  the  other.  [th.  5„ 

E.g.,     y/M^a^h"^  =  i/{Sa'b^  •  Qb)  =  2ab  {/6b. 

PrOB.  2.  To  FREE  A  RADICAL  FROM  COEFFICIE^^TS. 

Raise  the  coefficient  to  a  power  whose  exjjonent  is  the  7'oot-index 

of  the  radical; 
multiply  this  jjoiuer  by  the  expression  under  the  radical  sign, 

and  put  the  sa7ne  radical  sign  over  the  product,   [th.  5. 

E.g.,    2^5  ^6b  =  ^{Sa'b^-6b)  =  i^48a^K 

So,       a  ^(b^  -  c^)  =  ^[a''  -  (b^^  -  c") J  =  i^(«^Z»^"  -  aH^') . 

PrOB.  3.    To   REDUCE    RADICALS   TO   THE   SAME   DEGREE. 

Write  the  radicals  as  fraction  poivers;  [df.  fract.  pwr 

reduce  the  fraction  expo7ie7its  to  equivalent  fractions  having  a 

connnon  denominator;  '      [th.  3. 

restore  the  radical  sights,  using  the  common  de?io7ni?iator  as  the 

root'i7idex  a7id  the  7iew  n7i7nerators  as  expo7ients, 

E.g.,    ax,     ^by,     ^{b  +  c)=ax,  (byf^,        (b  +  cy^', 

=  («a;)«'/^     (byY'/'^      (b  +  cY^'^, 

=  ^(«.Tr,    Wyr.      'i^ih-hcY. 

PrOB.  4.    To  ADD   RADICALS. 

Reduce  the  radicals  to  their  simjilest  for7n;  [pr.  1. 

add  like  radicals  by  prefixi7ig  the  su7n  of  their  coefficie7its  to 
the  co7nmon  radical  factor ;  [add.  iiicom.  num. 

write  U7ilike  radicals  in  any  co7ivenient  order. 
E.g.,    34/8  +  5  1/2-104/32  =  64/2  +  5  1/2-404/2= -29  4/2. 
So,       a^b-^  a^  ^/b'  -  a'  ^b'  ^a'^b  +  a^b  ^b  -  a'b'  {/b 
z^(a-^a'b-a'b')i/b. 


1-4,  §3]  RADICALS.  173 

QUESTIONS. 

Reduce  the  radicals  below  to  their  simplest  forms: 

1.  125^/1  2.  567^'^  3.  392^1  4.  lOOS^/^. 

5.  216^/^  6.  721/^  7.,  162i/^  8.  48^ 

9.  1601/^  10.  (IIH)'''.      11.  (6fi)^/l  12.  (10^)^ 

13.  2500^         14.   1^296352.     15.   ^U^x-'yz^    16.  .y/bGa'lj%^. 

17.  ^n2a-'b-'cK  18.  -^64<f»<^-V.  19.  t'lG^^^^Vl 

20.  i/bOa'b'c'cF.    21.  i  |/(3/7).     22.  (a  +  b)  ^[{a-b)/(a-^b)l 

23    •'>4/     ^^'         24    ^^y-'^'       25    ^7   ^' 

26.  (r,'/Z.)  f/C^'^+Vft^-^).  27.  («-i/V46-2)-2.  28.  ^{x-^/^y'^Y* 
29.  V('2«'^-'^2Z> +  18^^-2^).  30.  l/[x'y-^-xy^-Zx%x-y)]. 
31.  3  |/U7-|  1^(1/3)-  i/(V-")-     32.  5  4/24-2  ^54-  y^6. 

rt-6    /^^^+2^^J^  r^^-^>^  /3aH6dZ^>  +  3/^^ 

^^-  .^_,yy  «  +  ^,         •         '^■^-    a  +  bV      9(a'-b')     ' 

Free  the  radicals  below  from  coefficients: 
35.  6  |/5.  36.  2  ^x,  37.  2.r  4/2.  38.  4a  |/5J. 

39.  4  >^6.  40.  Da  ^y.  41.  |  ^9'\  42.  i- 1/2^. 

._    a-b/a-hb         ..    o^y^lr>V'         ^^    3/       Z*^^/^ 

46.  al/a-WxYM'    47.  (:f2-?/2)3^.  |/[(:?-^)/(2:'  +  2:i'y  +  /)]. 

Reduce  the  radicals  below  to  the  same  degree: 

48.  a}'\  ci}^\         49.  «i/^  b^^K        50.  S^/^,  4^/*.        51.  a^/^  i^/a^ 

52.  ^ab,  ^ac,  ^bc,  i/{b  +  c).  53.  x^^^  a^/^,  a^^',  a^^',  3^/\ 

54.  3:^1/^  22/2/^  4;23/S  6v'^.         55.  y'rt^   ^6^   '^c^  "^^d",  ^eK 

Which  is  the  larger: 

56.  (l/2y/2  or  (2/3)2/3?     57.  ^^2   or    -^3?     58.   -^9  or    v^l8? 

59.  a)^/^  (\y^^,{iy^(.\Y^'^  so.  4/(«  +  ^  +  c)  or^/a-h  4/J+  4/c? 

Find  the  sum  of: 

61.    4/18-  y8.  62.  6|/(3/4)-3  |/(4/3). 

63.  3  4/(2/5)  +  4  |/(1/10).  64.  2  ^(1/5)  +  3  |^(l/40). 

65.    |/128  -  2  |/50  +  7  ^72.  66.  r/^^^/^  +  2a¥/^  +  ^'/^ 

67.  9  ^80  -  2  4/125  -  5  4/245  +  y  320. 


174  POWERS  AND  ROOTS  [VI,Prs. 

PrOB.  5.    To   MULTIPLY   RADICALS. 

If  all  the  radicals  have  the  same  base,  add  the  fraction  ex- 
ponents of  the  factors  for  the  exponent  of  the  product. 
If  the  bases  he  different,  but  the  radicals  be  of  the  same  degree, 

tvriie  their  product  under  the  common  radical  sign. 
If  the  radicals  be  not  of  the  same  degree,  make  them  like. 
If  there  be  coefficients, 'prefix  their  product  to  thcit  of  the  radical 
factors.  [V,  tli.  10. 

E.g.,    34/8.5  |/2.-10|/32=-3-5.10-  |/(8-2.32) 

=  -150.  |/512=-2400.  4/2. 
So,       ab''/^ '  a'^b'^^  'a-^b-y^=a'^^^-^'b''/^^^^^-y^=b-^/\ 

Note  1.  The  product  and  quotient  of  two  conformable  sim- 
ple quadratic  surds  are  rational;  of  two  such  non-conformable 
surds  they  are  surds. 

For  in  the  first  case,  the  surd  factors  occur  in  pairs  in  the 
product  and  vanish  in  the  quotient;  and  in  the  other,  they 
are  single,  and  the  square  root  cannot  be  taken. 
E.g.,    |/6    is  conformable  with     |/(2/3),    but  not  with     4/5, 
and       |/(6 .2/3),      |/(6 :  2/3),      4/(2/3 :  6)     are  rational, 
but        4/(6.5),      |/(6:5),      -^^(5:2/3)     are  surds. 

Note  2.  The  square  of  a  binomial  quadratic  surd  is  a  surd. 
E.g.,  if      4/«,      4/^     be  non-conformable  surds, 
then  is  (  4/^+  \/bf  =  a  +  b  +  2  \/ab,     a  surd. 

PrOB.  6.   To   GET   A   POWER    (OR  ROOT)    OF   A   RADICAL. 

Multiply  the  exponent  of  the  give^i  radical  by  the  exponent  of 
the  power  sought.  [th.  2. 

E.g.,    (3 .  8^/2)^     =  27  •  8^/'^  =  432  •  2^/^  ^  432  y2. 

So,       ^(3.  4/8)  =  p^72    =  ^72; 

(a^.  ^b'Y  =  a'^'  i/b^'=  a^^'b"'  \/b; 

Two  quadratic  binomial  surds  are  conjugate  if  they  diifer 
only  in  tlie  sign  of  one  term. 
E.g.,     rt+  \/h,a-  \/b]  10^/^3,  10^/2_3.   ^^^  ^y^  ^^_  ^^. 


5,6,  §3]  RADICALS. 


175 


QUESTIONS. 

Eeduce  the  expressions  below  to  their  simplest  forms: 
1.  3|/2.2|/3.  2.  84/G:2|/2.  3.  5  |/7-2  |/7. 

4.  3^'  =^ .  2^/1  5.  4  >t/3 . 3  |/5  •  5  p2.      6.  2^/^ .  31/3 .  41/*^ 

9.  3  yO .  2  i/3 . 4  |/5 :  12  ^10.      10.   ^V  •  (f )^'' :  (*)  •  (\W' 
11.  5^/*.42/^-33/2:60^/^  12.  (t^ -(l)'^':  A(l)"'^'- 

13.  |/(«2-^2):  j/(a-J):>^(fl-^').        14.  (5  +  2  4/2). (5 -2  4/2). 
15.  ia^b'-^bi/d'-ia-'/'b-'^'.  16.   (8  ^2  +  2  ^3)  •  (2  |/2  +  4/3). 

17.  (4+  |/2)-(l-  l/3)-(4-  4/2)-(5-  V3)-(l+  |/3).(5+  |/3). 

18.  («  +  ^)i/"^ .  («  +  ^)^^"  •  {a  -  ly^"^ '  [a  -  by  •  («'  +  Z.2)(»i+n)/mn 

19.  4/-rt-  V-Z*-  \/-a'  ^-b'  iZ-a-  ^-b-  ^/-a-  ^-b. 

20.  (4/2+  |/3+  4/5). (-  4/2+  V3+  V5) 

.(4/2-  |/3+  |/5).(>/2+  |/3-  4/5). 
Find  the  powers  and  roots : 

21.  (3  |/3)^  22.  (2  |/5)^  23.  (  4/2  -  4/3)^ 
24.  (4/10-  4/5)1  25.  (3^-^-3-1/2)2.  26.  (4/f-  4/!)^. 
27.  (21/3-2-2/3)3.       28.  (31/3-3-1/2)1        29.  (4i/3  +  4-i/y. 

30.  Pdx-iyy.         31.  [«3j(«3jc)i/5]i/«.     32.  |^-2i/"'«"*Z»"»c2m^ 
33.  («i/3^-i/3  +  «-i/3^i/3)3.  34.  [(«  +  Z>)V2_(^_j)i/2]2. 

35.  Find  the  first  five  powers  of  4/-I. 

Are  the  even  powers  of  \/~l  real  or  imaginary?  are  the 
fourth,   eighth,    twelfth   powers   positive   or   negative?    the 
second,  sixth,  tenth  powers  ? 
Write  the  surds  that  are  conjugate  to : 

36.  3-2  4/a:.        37.  4/3-24/5'.        38.  i/{a-\-b)-h  i/(a-b), 
39.3  +  24/5.    40.4/3  +  24/5.     41.  a;-3?/-i/2.     42.  a:i/2-3i/-i/2. 

43.  What  terms  of  the  square  of      4/^^+  \/b     are  products  of 

conformable  factors  ?  of  non-conformable  factors  ? 

44.  If  the  product  of  four  simple  quadratic  surds  be  rational, 

the  product  of  any  two  of  them  is  conformable  with 
that  of  the  other  two. 


176  POWERS  AND  ROOTS.  [VI,Pr. 

Two  surds  are  complementary  if  their  product  be  rational. 
E.g.,    a}/\     «2/^;       b^/^y     5-2/^;       ^{0^  +  1%     i/(«2  +  ^'^). 

So,  any  two  conjugate  binomial  surds  are  complementary; 
and  three  or  more  surds  whose  product  is  rational  form  a 
group  of  coynplementary  surds. 
Kg.,    a-\-yfh,     a-i/b;      2  +  3|/-l,    2^3j/-l; 
h-^{c-{-\^d),     i  +  i/ic  +  ^d),    b^-c  +  i^d. 

PrOB.  7.  To  REDUCE  A  FRACTION  WITH  A  SURD  DENOMI- 
NATOR TO  AN  EQUIVALENT  FRACTION  WITH  A  RATIONAL 
DENOMINATOR. 

(a)  The  denoniiuator  a  mono7nial:  multiply  both  terms  of  the 
fraction  by  some  complement  of  the  denomi^iator. 

{b)  The  de?iO)ninator  a  simple  bi^iomial  quadratic  surd:  muU 
tiply  both  terms  of  the  fraction  by  the  conjugate  of  the 
deno^ninator. 

E.g.,    a/W/'  =  a'b^/ybi    a/(^b-i/c)  =  a'{i/b  +  i/c)/(b-c). 

(c)  The  denominator  a  binomial  quadratic  surd  containing  a 

complex  radical :  multiply  both  terms  of  the  fraction  by 
a  group  of  conjugate  radicals  that,  taken  together,  are 
complementary  to  the  denominator, 

a        _«•  4/(^  +  4/6')  _ft- |/(Z'  +  ^c)-(^  — 4/6') 
'^■'     V(^  +  |/c)  ~       h^j^G~~~  W^c  ■• 

(d)  The  denominator  any  binomial  surd:  multiply  the  ttvo 

fraction  expo7ie?its  of  the  denominator  by  the  lowest 
conwion  multiple  of  their  denominators,  and  attach  the 
products  as  exponents  to  the  tico  bases; 

divide  the  sum,  or  difference,  of  the  poiuers  so  found  by  the  de- 
nominator, and  rmdtiply  the  mimerator  by  the  quotient. 

E.g.,  to  rationalize  the  denominator  of     6^/7(22/^3^/*) : 

then  •.'  12  is  the  lowest  common  multiple  of  3,  4, 

and       12.2/3  =  8,     12-3/4  =  9, 

and  v(28-3«):  (2^/^  +  3^/*) 

_  22a/3  _  220/3 ,  53/4  I  018/3 .  38/4  _  ,  ,  ,    I  02/3 .  330/4  _  333/4 

.-.  6^/^/(2^/^  +  3'/*)  =  G^/=* .  (2^^/^ 33^/*)/(2«  -  3^). 


7,  §3]  RADICALS.  177 

QUESTIONS. 

Prove  these  surds  complementary: 

1.  &^\&'\     2.  |/5-6,  4/5  +  6.    3.  x^'^^'K    4.  f^^y-'^K 
5.  2x-y\^-l,    2x  +  y^-l,        6.  a-\-b-^c,  a-]-b-\-\/c. 

7.  (4:-3i/x)/(5  +  6i/2),     (4  +  3V^)/(5-6|/2). 

8.  >j/2  +  |/3-|/5,     ^2  + 4/3  +  1/5,     |/6. 

9.  |/« - 4/^ - |/t^,     -f/rt - 4/ J  + -/c,  a  +  5  — c  +  24/«J. 

10.  The  product  of  any  two  conjugate  quadratic  binomial 
surds  is  rational:  what  other  complement  has  such  a  surd 
besides  its  conjugate  ? 

11.  The  product  of  two  surds  differing  only  in  the  sign  of 
one  term,  but  of  higher  degree  than  the  second,  is  not  rational. 

"Write  three  complements  of  each  of  the  surds: 

12.  7?i-3/2.        13.  (xY^.        14.  (a-by^K      15.  ab-^^\ 
Reduce  to  equivalent  fractions  with  rational  denominators: 

16.  1/4/3.        17.  6/^2.         18.  3|/8/24/2.    19.  2x/3y^/\ 
20.  4x/3f^K  21.  (a/yy^\     22.  (m/?i)-^^\    23.  2/(^/3  +  1). 

4^2-1  1/3-4/2  21  1  +  V5 

"      ^"2  +  1*  *  t^3  +  4/2*       ■  4/10-^3*  4/5-1/3* 

•  (a  +  ^y'^-(a-^)V2-  '*^-  «  +  ^[^  +  y(c+4/,/)]- 

4/2 -(1/2  ^3)  ^  +  (^-1)^-^^ 

^-  (4/2  +  8). (|/3-4/5)'  ''^'  ^-(a;2_i)i/3- 

a: -(3.-2 -1)1/2  ic-O'c^  +  l)^/^ 


1/2' 


•  («  +  4/-l)2-(«-4/-l)2'         •     4/(1 -^^)  + 4/(1  -  a^)  ' 

4/6-4/5-4/3  +  4/2  1  +  34/2-24/3 

*  4/6  +  1/5-4/3-4/2*  ^*    4/2  +  -^3  +  .v/6* 

38.  (3  +  >/3). (3  + 4/5). (V5~2)/(5-^5)/(l  + 4/3)^^4/15. 


178  POWERS   AND   ROOTS.  [VI, 

EQUATIONS  THAT   CONTAIN    SURDS. 

An  equation  that  contains  surds  is  rationalized  when  it  is 
replaced  by  an  equivalent  equation  free  from  surds. 

E.g.,  the  equation     x=\/2,    i.e.,     x='\/2   or  x=y2, 
when  rationalized,  becomes    x^—2. 

So,  if    \/x-i-^y  =  Oi 

then    x  —  y  =  0;  [mult.  ea.  mem.  by  j^x  —  4/y. 

or         V^^—Vl/f  [ax.  add. 

and      x=y.  [squaring. 
So,  if    4/ic+Vy  +  |/2!  =  0; 

then     x  +  y  —  z-\-2^xy  =  0,  [multhy  (^x  +  ^y-j/z), 

2  ^xy =z—x—y,  [ax.  add. 

and      4:xy  =  z^  +  a^+y^  +  2xy-2zx-2zy.  [squaring. 

So,  if  x'/'-{-x'/^-i  =  0: 

then     a;- 22:^/^  +  1  =  0,  [mult,  by  (2:^3-1). 

2x^^^  =  x-{-l,     8x  =  x^-\-3a^-i-3x  +  l,  [cubing, 

and      a^  +  3x^-bx-{-l  =  0. 

^'        x+i^x-Q       44/^4-1' 

ti^en    '^^~;z  =  -r-^ =-,        Fred,  first  mem.  to  lowest  terms. 

^x-2      4^x  +  l'        ^ 

Ax-15^x-4:  =  x  —  4:,  [clear,  of  f ract. 

and     x  =  5^x,     a^=25x,    a;  =  25orO, 
So,  if  i/{3+x)  +  i/x  =  Q/^(^  +  x): 
then    3  +  a;  + 1/(3:?;  +  (^)  =  Q,  [clear  of  f ract, 

i/{3x  +  a^)  =  3-x, 
and      3a:  +  a;'*=9-6:r  +  a;^     9a;  =9,     a:=:l.  [squaring. 

So,  if  x'^-[x-(l-xY^J^'=l: 
then     [^  -  (1  -  ^Y^'Y^  =  x'^''  - 1,     a:  -  (1  -  3:)*/^  =  ic  -  2x'/^  + 1, 

(1  -  ic)^/«  =  2:r^/=^  - 1,     (1  -  a:)  =  42:  -  4a:^/«  + 1, 
and      4x^/^  =  52:,     l^x  =  2ba^,     a:  =16/25    or  0. 


§3]  RADICALS.  179 

QUESTIONS. 

Solve  the  equations : 

1.  -^/(3-.t)  +  6  =  7.    2.  |/(:,;  +  4)  =  4-4/:c.      3.   ?/»/H5  =  ll. 
4.  x-^a  =  \/[a''  +  x\/{P-\-x')].  5.  |/(20-|/2:r)=:2. 

6.    — ,— J-  =  4-1 .    [red.  first  f ract.  to  lower  terms, 

\/ax  +  l  2  "- 

|/3.r-|/3^a:  +  3  4/(42^  +  1) +  |/4r 

•  |/2x--4/2         2    •  •  i/{^x  +  l)-y^4:x  ~^' 

9.  3^^^+{a  +  xy/^  =  2a{a+x)-^^^  10.  (2:2-iy/2=8(a;2-l)-V2. 
11.  (a^-3a;  +  4)V2=a:-3.        12.   {8-4:xY/'+{ld-^xY/'=5. 

13.  18(2^  +  3)-i/2=(2a;-3y/H(22:  +  3)i/2. 

14.  3|/(.T-f) +74/(2-  +  /^)  =  10|/(a;  +  .03). 

15.  ^(|/3  +  ayr)  +  t'(4/3-.r|/7)rz:^12. 

16.  |/(r;  +  x)  + 1/(«  -  x)  =  ^;[|/(rt  +  a-)  -  ^(a  - :?:)]. 
Rationalize  the  equations: 

17.  ^a  +  i/b  +  c  =  0.  18.  3|/a:-2|/5j/=-l. 
19.  |/a:-|/j/  =  |/y-|/a;.  20.  -  l^a;^  _  ^^  ^  ;i^  _  q^ 
21.  ^a^-{/x-l  =  0.                    22.  ^x''-^x-\-l  =  0. 

23.  What  effect  is  produced  on  the  degree  of  an  equation 
by  rationalizing  it? 

24.  Divide  18  into  two  parts  whose  squares  shall  be  to  each 
other  as  25  to  16.  In  solving  this  problem  four  square  roots 
are  found :  what  is  the  effect  if  one  of  them  be  taken  negative  ? 

25.  If  to  a  certain  number  22577  be  added,  the  square  root 
of  the  sum  be  taken,  and  from  this  root  163  be  subtracted, 
the  remainder  is  237:  what  is  the  number? 

2G.  The  length  of  the  side  of  a  square  whose  area  is  1  square 
inch  less  than  that  of  a  given  square,  increased  by  the  side  of 
a  squtire  whose  area  is  4  square  inches  more  than  that  of  tlie 
given  square,  equals  the  side  of  a  square  whose  area  is  5  square 
inches  more  than  4  times  that  of  the  given  square;  find  the 
area  of  the  given  square. 


180  POWERS   AND   ROOTS.  IVI.  Pr. 

§4.   ROOTS   OF  POLYNOMIALS. 

Evolution  is  an  inverse  operation :  the  work  is  an  ejffort  to 
retrace  the  steps  taken  in  getting  the  power  whose  root  is  now 
sought;  it  is  a  process  of  trial,  by  progressive  steps,  like  divi- 
sion and  other  inverse  operations,  and  its  success  is  established 
by  raising  the  root  to  the  required  power  and  finding  the 
result  identical  with  the  given  polynomial. 

SQUARE   ROOT. 
PrOB.  8.   To  FIND   THE   SQUARE   ROOT  OF  A   POLYIfOMIAL. 

Arrange  the  terms  of  the  polynomial  in  the  order  of  the  poiuers 

of  some  one  letter,  a  perfect  square  first; 
tahe  the  square  root  of  th  first  term  of  the  polynomial  as  the 

first  tenn  of  the  root; 
divide  the  remainder  ly  double  the  root  so  found,  and  make 

the  quotient  the  next  term  of  the  root  and  of  the  divisor; 
multi2:)ly  the  complete  divisor  by  this  term  and  subtract  the 

product  from  the  divideyid; 
double  the  root  found  for  a  new  trial  divisor,  and  proceed  as 

before. 
The  rule  is  based  on  the  type  form  for  the  expansion  of  the 
square  of  a  polynomial;  when  any  complete  divisor  is  mul- 
tiplied by  the  new  term  of  the  root,  and  the  product  is  sub- 
tracted from  the  last  remainder,  the  whole  root  thus  far  found 
is  thereby  squared  and  subtracted  from  the  polynomial. 
E.g.,  rt'  +  2ab  +  Z/H  2ac  +  '^bc  ^-(^(a  +  b  +  c, 

'Za  +  b)2ab  +  y 


2a  +  2^  +  c  )2ac-¥2bc-\-c^ 
So,  (Zx-^-ay^/^  +  a-h*' 

Qx-^  -  ay^^')-Qax-y/^-\-aY/^ 

Qx-^-  iay^/^  +  a  - h*)  6a - ^x -  V  -  2y^^h*  +  a-^s^ 


8.  §4]  ROOTS   OF   POLYNOMIALS.  181 

QUESTIONS. 

Find  the  square  root,  preferably  by  detached  coefficients,  of: 
1.  163f-'i0xy  +  2DyK  2.  l  +  2x-^73^-\-Ga^  +  9xK 

3.  a*-2c^  +  2a''-a  +  i,  4.  x-*^-2x-^-\-Z-2x^  +  xK 

^    f    X      y     1  a'     2a  2h     W 

y'    XT    y      X    4:  Ir     o  a     or 

7.  9a;2-30fla;-3fl2a;  +  25flH5ft'  +  irt*. 

8.  4:a^-12ab  +  4ax  +  W-Qbx-{-x', 

9.  9««-12fl5-26a*  +  44«H9«2-40rt  +  16. 

10.  4^;* +  8aa;^  +  4fl^2^  + 166^2^ +  16fl^2_y^  16^4^ 

11.  ^(M  -  iabs^z + ^a'bx'z^  +  b^v'z'  -  4« J2^^  +  4a252:c2^. 

12.  4::>^-\-ix-^  +  lQx-'-^^16x-''-p-K 

14.  1  +  4/2:  +  lO/a^  +  20/^  +  25 /a^  +  24/af  +  16/a:«. 

15.  1  -  4/^  + 10/^2  -  20/z'  +  2d/z*  -  24/^^  4- 16/2^ 

16.  How  many  terms  has  the  square  of  a  binomial  ? 
Supply  a  third  term  to   a^-\-2ab   that  shall  make  the  result- 
ing trinomial  a  perfect  square;  so,  to     a^  +  W]  to     2ab  +  b'^. 

Make  a  general  rule  for  completing  such  a  square. 
Complete  a  square  from  each  of  the  expressions: 

17.  x'  +  Qx.         18.  uf-^x.  19.  x'  +  bx.        20.  x'-lx. 
21.  3^-Zx^.       22.  x^  +  Sx^.        23.  y''-12y\     24.  92^  +  12;^. 
25.  16a^-24a;.  26.  25a:*-20a;l  27.  92*-122^    28.  4.zr-102;. 
29.   (y^  +  y-Qf-^y^  +  y-Q).    30.   {x  +  yf +  2{x  +  y). 

31.  (a:2_^2a;-l)H42:2_^g^_4   32.  {a-^y  +  Q{a-^y)\ 
Find  four  terms  of  the  square  root  of : 

33.  1  +  7^.  34.  x'  +  l.  35.  u?-a\         36.  «''^-ar'. 

37.  l-a:^.  38.  ^-1.  39.  2;2  +  «'.         40.  a^  +  x". 

Show  that  the  expressions  below  are  perfect  squares: 

41.  (x^-yzY+{y^-zxy+{z^-xyY 

-3{:>^-yz)-iy'-zx).(z'-xy). 

42.  4[(a'-I^)cd-{-((f-(P)abY-^  [{a'-b')(o^-d')-4:abcd]\ 


182  POWERS  AND   ROOTS.  [VI.Pr 

CUBE   ROOT. 
PrOB.  9.    To   FIND   THE   CUBE   ROOT   OF  A   POLYNOMIAL. 

Arrange  the  terms  of  the  ^jolynomial  in  the  order  of  thepoiuers 

of  some  one  letter,  a  perfect  cube  first; 
take  the  cube  root  of  the  first  term  of  the  polynomial  as  the  first 

term  of  the  root; 
divide  the  first  term  of  the  remainder  by  three  times  the  sq^iare 

of  the  root  found,  and  make  the  quotient  the  next  term 

of  the  root; 
to  the  divisor  add  three  times  the  product  of  the  first  term  of 

the  root  by  the  second  term,  and  the  square  of  the 

second  term; 
multiply  the  complete  divisor  by  this  term  and  subtract  the 

product  from  tlie  dividend; 
take  three  times  the  square  of  the  root  found  for  a  new  trial 

divisor  and  proceed  as  before,  treating  the  root  so  far 

fo2i7id  as  the  first  term  and  the  new  term,  as  the  second. 
The  rule  is  based  on  the  type-form  for  the  expansion  of  the 
cube  of  a  polynomial 

{a  +  bY  =  a^  +  ^a^b  +  ?.ab^  +  b''  =  a^+{'^a^  +  ?>ab  +  W)b', 
and  when  any  complete  divisor  is  multiplied  by  the  new  term 
of  the  root  and  subtracted  from  the  remainder,  the  whole  root 
so  far  found  is  thereby  cubed  and  subtracted  from  the  poly- 
nomial; 3a^  is  the  trial,  ?fa^-\-'6ab  +  b^  the  complete,  divisor. 
E.g.,         {a-\-h  4-  c 

/»3 


3^2  +  3«6  +  b^ )   2>a'^h  +  3a6^  +  &' 

3^2  J\.  Qah  +  36^+  3ac  +  ^hc   +  c^     )3tf.'^c  +  6a6c+3acH3^>-c  +  dbc''-\-c^ 

The  rule  given  and  illustrated  under  prob.  11  is  perhaps 
the  better  rule  for  getting  cube  roots;  the  cubing  of  tlio  root 
takes  the  place  of  the  less  familiar  process  of  completing  the 
divisor. 


9,§4] 


ROOTS  OF  POLYNOMIALS. 


183 


QUESTIONS. 

Find  the  cube  root,  preferably  by  detached  coefficients,  of; 


1.  l  +  6.r  +  12a;2  +  8a:^. 


9, 


+  «■ 


2.  x^-6x^i/  +  12x''y^-8y\ 
4.  x^y-^-Qx^ +  12x^1/^ -8i/\ 

'^2(^af^4:(^a^'^8c'x'' 
-6T'-\-iO:(^-dQx-64:, 


3^       1  .. 

'  8  "^2  '^3x^'^27x^' 

7.  a;H6.r^- 40^:^  +  96.?; -64.      8. 

9.  «^-9a*^5c  +  27rt2^V-27^V. 

10.  l-6a;  +  21a;2-44.^+63a:*-54.7/  +  27a;®. 

11.  {a  +  iy''x^-6caP{a  +  iy\(f-hl2(fa'P{a-\-lY''x-8(^a^. 

12.  Sa;^  +  483^y  +  60:cy -  SO^-^i/^ -  90^'^  +  lOSa:^^ _  2^^^^ 

13.  ic3-9a;2  +  30a;-45  +  30a:-i-9a;-2  +  2:-l 

14.  a^/*  -  Grt^/*  -  9«3/2  _f.  i2ayi  +  36«H  19a«/*  -  36fl5/2  _  54^11/4 


Find  the  cube  root,  preferably  by  the  type-form,  of: 


.  a^-Gy   x^  +  12y^ 

X-   8/ 

a^  +  6y 

^^  +  12y' 

x-h   8y' 

-\-9z       -d6yz 

+  36yh 

-9z 

-3Qyz 

-36yh 

+  27^2 

-D4:yz' 

'                    +27^2 

+  54^/2^ 

+  272^ 

-27^ 

16.  Arrange  the  terms  in  ex.  15  to  rising  powers  of  y  and  then 

find  the  cube  roots;  so,  to  falling  powers  of  z. 
In  what  are  these  cube  roots  alike,  and  how  do  they  differ? 

17.  a^-\-6a'b  +  12aP  +  8P  18.  a^  +  Sx^y  +  Uxif  +  Sf 
-a'  +  Qa'b-12ab^  +  8b^  -a^  +  6a^y-12xy"  +  8f 

8a]i-12a'b  +  6ab^  +  i\  Sa^-hUx'y -h6xy'  +  f. 

19.  In  the  cube  of     a-\-b,     how  many  terms  are  of  the  form 

a^  ?  of  the  form  3a^b  ?  how  many  terms  in  all  ? 

Without  finding  the  root,  show  that     x^-\-Q.'):^y-\-12xy^  +  8y^ 

and     x^  —  6x'^y  +  12xy^  —  8y^     are  perfect  cubes. 

20.  Find  the  cube  of     a  +  b  +  c.     How  many  terms  are  of  the 

form  a^?  of  the  form  3a^b?  of  the  form  (^abc?  in  all? 
Can  a  polynomial  of  two  terms  be  a  perfect  cube  ?  of  three 
terms?  of  four?  of  five  ?  of  eight  ?  of  nine  ?  of  ten? 


184  POWERS   AND  ROOTS.  [VI,  Prs. 

PrOB.    10.     To    FIND   THE   ROOT   OF   A   POLYNOMIAL,   WHEN 
THE   ROOT-INDEX   IS   A   COMPOSITE   NUMBER. 

Resolve  the  root-index  into  its  prime  factors; 

find  that  root  of  the  polynomial  whose  index  is  the  smallest  of 

these  factors;  [th.  2. 

of  this  root  find  that  root  tvhose  index  is  the  7iext  larger  factor, 

and  so  on. 
E.g.,    the  fourth  root  is  the  square  root  of  the  square  root; 

the  sixth    root  is  the  cube     root  of  the  square  root; 

the  eighth  root  is  the  square  root  of  the  square  root  of 
the  square  root. 

So,       ^{x^-Sxl+^Sx^-b^x'  +  lO-fiQx-^+^Sx-^-Sx-^+x-^) 
=  x  —  x~^. 

PrOB.    11.    To    FIND    THE    7iTH    ROOT    OF    A    POLYNOMIAL 
WHEN   n   IS   PRIME. 

Arrange  the  terms  of  the  polynomial  in  the  order  of  the  powers 
of  some  one  letter,  a  perfect  nth  power  first; 

write  the  nth  root  of  the  first  term  as  the  first  term  of  the  root; 

divide  the  second  term  ly  n  times  the  (7i  —  l)th  power  of  the 
first  term  of  the  root,  as  a  trial  divisor,  and  make  the 
quotient  the  second  term  of  the  root;  [th.  X 

from  the  given  polynomial  suUract  the  nth  power  of  the  bi- 
nomial root  already  found; 

divide  the  first  term  of  the  remainder  hy  the  same  trial  divisor 
as  before,  and  so  continue,  ahvays  subtracting  the  nth 
poioer  of  the  root  so  far  found  from  the  entire  polynomial. 

E.g. ,    xy^l  -  xyz  +  2X-1  +  IS/x  -  27/a;2  +  27/ar>(a:/3  - 1  +  3/a; 
3?/21-xy^+   x-1 
a:V3)a:-6 
ar'/27  -  ar'/S  +  2:^  -  7  +  IS/x  -  27/ar^  +  27/«». 


10,11,5  4]  ROOTS   OF   POLYNOMIALS.  185 

QUESTION'S. 

Find  the  fourth  root  of : 

1.  .T«  -  4:x'  +  10.?;«  -  16rH  19a;*  -  16a;^  +  lOrc^  -  4a:  + 1. 

2.  a:*  +  4a:3^  ^  6^2^2  ^  ^^^s  _f_  ^4  _^  4^^  ^_  j  Ox^yz  +  ISa:?/^;^ 

+  4  ?/2  +  exh""  +  12xyz^  +  6fz^  +  4a;z3  +  4?/23  +  zK 

3.  Expand  (r^  +  ^)*.     How  many  terms  of  the  form  a*?  of 

the  form  ia^b  ?  of  the  form  Qa^b^?  in  all  ? 

4.  Expand   {a-\-b-hc-{-dy.     How  many  terms  of  the  form 

a*  ?  of  the  form  4:(fb  ?   of  the  form  6a^b'^  ?  of  the  form 
12a^bc?  of  the  form  2iabc(l?  in  all  ? 

5.  Can  an  expression  of  four   terms  be   a  perfect  fourth 

power  ?  of  five  terms  ?  of  nine?  of  ten?  of  fifteen  ? 
Find  the  sixth  root  of: 

6.  «^-12rt*  +  60rt2_l60  +  240ft-2-192«-H64«-^ 

7.  a'  -  2a'b  +  ^a'b'  -  \^a^W  +  -i-^a%'  -  -ijCib'  +  jyb\ 

8.  x"^  - 12.^1*^  +  6Qx^  -  220a:«  +  495a;*  -  T92a;2  _,_  924  _  '::q2x-^ 

+  4:9dx-^-220x-'  +  66x-^-12x-^^  +  x-^. 
By  the  process  of  prob.  11  find  the  cube  root  of: 

9.  a'  -  6a'b  +  loa'b' -  20a^^  +  loa'b' -  Qab'  +  b\ 

10.  a;3  -  8/  +  21  z^  -  6.^y  +  12a;^2  +  9xh  +  27xz^  4-  36 fz  -  5iyz* 

-3Qxyz. 
Find  the  fourth  root  of: 

11.  a;*-12a;3^62a,'2-180^  +  321--360a;-i  +  248a;-2-96a;-3 

+  16a;-*. 

12.  (a;Ha;-2)2_4(a;  +  a;-i)H13. 
Find  the  fifth  root  of: 

13.  a'  +  lOa'b  +  4.0a^J  +  80a^^  +  SOa¥  +  326^  -  5a*6'  -  ^Ocv'bc 

-  120a'b~c  -  IQOaWc  -  SCb^o  +  lOaV  +  60a'b(f  +  120abY 
+  80^  V  -  lOrt  V  -  4:0ab(^  -  40^»  V  +  5ac^  +  lO^c*  -  c^. 
Find  the  sixth  root  of: 

14.  64a;«-384a;*  +  9G0./;2-1280  +  960a:-2-384a;-*  +  64a:-«. 
Find  these  roots  correct  to  three  terms: 

15.  v'Cl-Sa;).    16.  -^(l-Sa;^).    17.  v'(l-4a;^).   18.  f(l-5a;*). 


186  POWERS   AND   ROOTS.  [Vl.Pa. 

§5.   ROOTS   OF  NUMERALS. 

PrOB.  12.   To   FIKD   THE   SQUARE   ROOT   OF  A   NUMERAL. 

Separate  the  mineral  into  periods  of  two  figures  each,  both  to 

the  left  and  to  the  right  of  the  decimal  point; 
take  the  square  root  of  the  largest  perfect  square  in  the  left- 
hand  period; 
subtract  this  square  from  the  period,  and  to  the  remainder 

annex  the  next  period  to  form  the  first  dividend; 
double  the  root  already  found  and  use  it  as  a  trial  divisor, 

omitting  the  last  figure  of  the  dividend; 
annex  the  quotient  to  the  root  and  to  the  trial  divisor; 
multiply  the  complete  divisor  by  this  root  figure,  subtract  the 
product  from  the  divideiid,  bring  down  the  next  period, 
and  proceed  as  before. 
Note   1.   Numerals  are   polynomials,  but   polynomials  in 
which  the  terms  overlie  and  hide  each  other;  and  virtually 
the  rule  for  finding  roots  is  the  same  for  both. 

The  separation  into  periods  is  a  matter  of  convenience  only; 
it  comes  from  this :  that  the  figures  of  the  root,  of  different 
orders,  are  best  found  separately,  and  that,  since  the  square  of 
even  tens  has  two  O's,  therefore  the  first  two  figures,  counting 
from  the  decimal  point  to  the  left,  are  of  no  avail  in  getting 
the  tens  of  the  root,  and  are  set  aside  and  reserved  till  wanted 
in  getting  the  units'  figure.  So  the  square  of  even  hundreds 
has  four  O's,  and  the  first  four  figures,  two  periods,  are  set 
aside  and  reserved  till  wanted  in  getting  the  tens;  and  so  on. 
So,  in  getting  roots  of  decimal  fractions,  the  square  of 
tenths  has  two  decimal  figures,  and  the  first  two  figures,  one 
period,  are  used  in  getting  the  tenths'  figure  of  the  root;  the 
square  of  hundredths  has  four  decimal  figures,  and  so  on. 
The  same  thing  appears  from  this:  that  the  root  of  a  fraction 
is  most  easily  found  if  the  denominator  be  a  perfect  square; 
and  this  it  is,  with  decimal  fractions,  only  when  it  consists  of 
1  with  two  O's,  or  four  O's,  or  six  O's,  and  so  on;  that  is,  when 
the  number  of  decimal  figures  used  is  even. 


12,  §5]  ROOTS   OF  NUMERALS.  187 

QUESTI01T8. 

Find  the  square  root  of : 

1.  494209.  2.  3345241.  3.  125457.64. 

4.  3533.1136.  5.  17.75358225.  6.  11090466. 

7.  1732.323601.  8.  576864324.  9.  1771561. 

Find  the  square  root,  correct  to  three  decimal  places,  of: 
10.  144.  11.  14.4.  12„  1.44.  13.  .144. 

14.  .0144.       15.  .00144.      16.  .000144.        17.  .0000144. 
Find  the  fourth  root,  correct  to  three  decimal  places,  of; 

18.  16.0001.    19.  160.001,    20.  1600.01.        21.  16000.1. 
Find  the  square  root,  correct  to  a  twelfth,  of: 

22.  49/144.     23.  49/72.       24.  49/36.  25.  49/18. 

Find  the  square  root,  correct  to  a  ninth,  of: 

26.  17/9.    27.  17/27.   28.  17/6.     29.  17/36. 

30.  Find  the  square  of  15;  then  find  (10  +  5)^,  {d  +  Qf, 
using  the  type-form,  and  show  that  the  powers  and  products 
of,     10,  5,     9,  6,     overlie  and  hide  each  other  in  the  results. 

31.  In  getting  the  square  root  of  225,  show  that  10  is  not 
necessarily  the  first  guess.  Make  the  computation,  using  9 
for  the  first  number;  then,  in  turn,  using  12,  16,  20,  for  the 
first  number. 

32.  How  many  figures  in  V  ?  in  9^  ?  in  10^  ?  in  99^  ?  in  the 
square  of  the  smallest  three-figure  number?  the  largest? 

How  many  figures  in  the  square  of  a  number  of  n  figures  ? 

33.  How  many  figures  are  there  in  the  square  root  of  an 
integer  of  four  figures?  of  three  figures?  of  2n  figures?  of 
2n  —  l     figures? 

34.  What  is  the  object  of  separating  a  numeral  into  two- 
figure  periods?  Why  could  not  the  process  begin  with  the 
first  figure  of  the  number,  irrespective  of  the  decimal  point? 

35.  Why  must  a  decimal  fraction  that  is  a  perfect  square 
be  expressed  by  an  even  number  of  figures? 

36.  How  many  decimal  figures  has  a  perfect  7ith  power  ? 


188 


POWERS  AND  ROOTS. 


[VI,  PR. 


GEOMETRIC   ILLUSTRATION. 

Note  2.  The  reason  of  the  rule  for  finding  the  square  root 
is  made  clear  by  a  geometric  illustration. 
E.g.,  to  find  the  side  of  a  square  of  6064  square  inches. 

The  length  of  the  side  lies  between  70  and  80  inches. 


B 

C 

A 

4900  sq.  in. 

B 

The  square  A,  70  inches  on  a  side,  contains  4900  square 
inches,  leaving  1184  square  inches,  to  be  so  added  that  the 
resulting  figure  shall  be  a  square.  This  is  done  by  adding 
equal  rectangles,  B,  on  two  sides  and  a  square,  c,  to  complete 
the  figure.  


70  in.  70  in. 

These  additions,  placed  end  to  end,  form  a  rectangle  wh(^se 
length  is  a  little  greater  than  140  inches,  and  whose  area  is 
1184  square  inches.  The  breadth  of  this  rectangle  is  a  little 
less  than  the  quotient  of  1184  by  140.  Try  8  inches:  then 
the  entire  length  of  b  +  b  +  c  is  148  inches,  and  the  area 
(148  X  8)  square  inches,  is  the  desired  addition. 

The  side  of  the  square  is  therefore  78  inches. 

CONTRACTION". 

Note  3.  When  the  first  7i  figures  of  a  square  root  have 
been  found  by  the  rule  above,  then  n  —  1  more  figures  maybe 
got  by  dividing  the  remainder  by  double  this  root. 

In  principle,  this  process  is  the  same  as  using  a  trial  divisor; 
only  the  possible  error  in  the  quotient  needs  consideration. 

The  whole  root  is  divided  into  two  parts/  the  first  }i  figures, 
already  found,  and  the  n  —  \  figures  that  follow. 
Put  p,  Q,  for  the  values  of  these  two  parts; 
then-/p>Q-10"-*, 


12,  §5]  ROOTS  OF   NUMERALS.  189 

.-.  2p  differs  from  2p  +  q,  the  complete  divisor,  by  less 
than  oue  part  in  2-10""^  parts,  of  this  divisor, 
and       the  resulting  quotient,  q',  differs  from  the  true  quo- 
tient, Q,  by  less  than  one  part  in  2  •  10""^  parts,  of  Q. 
And-.-Q<10"-S 

.*.  q'  is  in  error  by  less  than  half  a  unit  in  the  last  figure. 

QUESTIONS. 

By  contraction  find  the  value,  correct  to  three  decimal  places,  of: 

I.  |/185.      2.   4/912.       3.  ^729.      4.   ^1008.       5.  ^"8000. 

6.  If  the  square  root  of  a  number  contain  three  figures  and 
two  of  them  have  been  found,  the  part  of  the  root  still  to  be 
found  is  less  than  a  tenth  of  the  whole  root,  and  less  than  a 
twentieth  of  the  next  divisor. 

7.  The  quotient  found  by  using  a  divisor  between  twenty 
twenty-firsts  and  twenty-one  twenty-firsts  of  the  true  divisor 
gives  a  quotient  whose  error,  if  any,  is  less  than  a  twentieth 
of  the  remaining  figure.     Can  this  error  be  half  a  unit  ? 

8.  If  three  out  of  five  root  figui-es  have  been  found,  what 
part  of  the  whole  root  is  the  value  of  the  remaining  figures? 

The  error  in  the  contracted  method  is  less  than  what  frac- 
tion of  the  remainder  of  the  root  ?  what  fraction  of  a  unit  ? 

9.  If  n  figures  of  a  root  have  been  found,  what  is  the 
greatest  possible  error  that  can  be  introduced  by  finding  the 
next  n  —  1  figures  by  division  ? 

10.  If  a  true  quotient  be  27365.7  and  the  division  be  carried 
to  but  five  figures,  shall  the  fifth  figure  written  be  5  or  6  ? 

II.  If  the  true  root  be  273G5.7,  and  if,  after  finding  three 
figures,  the  next  two  be  got  by  division,  what  danger  is  there 
that  the  fifth  figure  be  written  5  and  not  6  ? 

If  the  true  root  be  27365.49,  what  is  the  danger? 

In  writing  down  the  last  figure  of  the  root,  in  which  direc- 
tion should  an  allowance  be  made  ? 

12.  Find  the  square  root  of  40297104,  and  illustrate  by  a 
diagram  showing  the  rectangular  additions  that  give  the 
hundreds'  figure,  the  tens'  figure,  and  the  units'  figure. 


190  POWERS   AND  ROOTS.  [VI.Pr. 

PrOB.  13.    To   FIND   THE   CUBE   ROOT   OF  A   NUMERAL. 

Separate  the  mtmeral  into  jjeriods  of  three  fig ^ires  each,  both  to 
the  left  and  to  the  right  of  the  decimal  poijit; 

take  the  cube  root  of  the  largest  perfect  cube  in  the  left-hand 
period; 

subtract  this  cube  from  the  period,  and  to  the  remainder  annex 
the  next  period  to  form  the  first  dividend; 

to  three  times  the  square  of  the  root  already  found  annex  tiuo 
ciphers  and  use  the  result  as  a  trial  divisor,  placing 
the  quotient  as  the  next  figure  of  the  root; 

to  the  trial  divisor  add  three  times  the  product  (with  one 
cipher  annexed)  of  the  first  part  of  the  root  by  the  new 
root  figure,  and  add  the  square  of  this  figure; 

multiply  the  complete  divisor  by  this  figure,  subtract  the  prod- 
uct from  the  dividend,  brhig  dowti  the  next  pe7'iod,  and 
proceed  as  before. 
Note  1.  The  reason  for  separating  the  number  into  periods 

of  three  figures  is  made  evident  by  considering  the  number  of 

figures  in  the  cubes  of  numbers  having  one  digit,  two  digits, 

three  digits,  and  so  on,  and  the  number  of  zeros  in  the  cubes 

of  exact  tens  and  hundreds.  [comp.  pr.  12  nt.  1. 

CONTRACTION. 

Note  2.  When  the  first  n  figures  of  a  cube  root  have  been 
found  as  above,  then  w  — 2  more  figures  may  be  got  by  divid- 
ing the  remainder  by  three  times  the  square  of  this  root. 
Put  p  for  the  value  of  the  figures  already  found,  and  Q  for 

that  of  the  ?^  — 2  figures  that  follow; 
then-.-  P>Q-10"-S 

.-.  3p^  the  trial  divisor,  differs  from     Sp^^  +  Spq  +  c^',    the 
complete  divisor,  by  less  than  one  part  in  10""^  parts, 
and       the  resulting  quotient,  q',  differs  from  the  true  quo- 
tient, Q,  by  less  tlian  one  part  in  10""'  parts,  of  Q. 
And-.-Q<10"-2, 

.*.  q'  is  in  error  by  less  than  a  tenth  in  the  last  figure. 


13,  §5]  ROOTS   OF   NUMERALS.  191 

If  the  first  figure  of  the  root  be  2,  or  larger,  then  7i  —  l 
figures  may  be  found  by  division. 

QUESTIONS. 

Find  the  cube  root  of: 

1.  148877.  2.  .007821346625. 

3.  2139656.927128.  4.  836.802326004904. 

Find  the  cube  root,  correct  to  three  decimal  places,  of: 

5.  1728.  6.   172.8.  7.   17.28.  8.  1.728. 

9.  .1728.         10.  .01728.         11.  .001728.       12.  .0001728. 

By  contraction,  find  the  value  correct  to  three  decimal  places,  of: 

13.  |/625.         14.  ^587.  15.  |/1728.  16.  ^^18.625. 

17.  How  many  figures  in  1^?  in  9^  ?  in  10^?  in  99^?  in  the 
cube  of  any  three-figure  number?  of  any  >^-figure  number? 

18.  How  many  figures  in  the  cube  root  of  a  number  ex- 
pressed by  on  figures?  3/^  —  1  figures?  3/^  —  2  figures  ? 

19.  If  three  out  of  four  figures  of  a  cube  root  have  been 
found;  if  this  part  be  called  a,  and  the  remaining  part  b;  then 
3ab-\-b^  is  the  part  of  the  divisor  omitted  in  the  contracted 
process,  b  is  smaller  than  a  hundredth  part  of  a,  3abi-b'^  is 
smaller  than  a  hundredth  part  of  the  true  divisor,  and  the 
error  in  the  quotient  is  smaller  than  a  tenth  part  of  a  unit. 

20.  If  262144  cubic  inches  are  to  be  arranged  in  the  form 
of  a  cube,  and  if  a  cube  be  formed  whose  edges  are  60  inches, 
how  many  cubic  inches  are  so  used  and  how  many  are  left  ? 

If  additions  60  inches  square  made  to  the  top,  front,  and 
one  side  face,  would  complete  a  perfect  cube,  how  thick  could 
these  additions  be  ? 

21.  Regarding  the  integer  part  of  the  result  in  ex.  20  as  the 
second  figure  of  the  root,  make  four  other  additions,  three  of 
them  with  a  length  of  60  inches,  and  a  width  and  thickness 
each  equal  to  the  thickness  of  the  first  additions. 

What  are  the  dimensions  of  the  final  addition  required  ? 
How  many  cubic  inches  have  thus  been  added  ? 
Draw  the  original  cube,  and  the  several  additions. 


192  POWERS   AND   ROOTS.  [VI,Ths.6,7. 

§6.   EOOTS   OF   BINOMIAL  SURDS. 

Theor.  6.  If  two  simjjle  surds  of  the  same  degree,  in  their 
simplest  forms,  he  equal,  their  coefficients  are  equal  a7id  their 
radical  parts  are  equal. 

Let  «f/A,  l\^'R  be  equal  simple  surds  in  their  simplest  forms; 
tlien  will     a^h,     A  =  B. 

For,     a:&=|/(B:A),     a  true   equation,  but  true  only  when 

^(b  :  a)     is  rational,  [the  5  cr. 

i.e.,      when     a  =  b;    and  in  that  case     a  =  l)    also.         q.e.d. 

Cor.  1.   T'wo  7ion-conformahle  si7iiple  surds  caimot  he  equal, 

Theor.  7.  Tlie  sumoftivo  simple  non-conformahle  quadratic 
surds  cannot  he  rational. 
For,  let  \/k,   \/B   be  two  simple  non-conformable  quadratic 

surds,  and  a,  h,  G,  be  rational  numbers, 
and  if  possible  let    a\/A  +  h\/B  —  c ; 

then  V  2ab'\/AB  =  G^  —  a^A  —  h^B,      [sqr.  both  mem.  and  transp. 
and      |/AB,     2ah\^AB,     are  surds,  [pr.  5,  nt.  1. 

,%  a  surd  equals  a  rational  number,  which  is  impossible, 
.\a^Ai-h\/B^G,  Q.E.D.  [df.  surd. 

Cor.  1.  If  a,  X  he  rational  numhers,  \/h,  4///  simple  quad- 
ratic surds,  and    x-\-^y  =  a-\-  \/h,     then    x  —  a,    \/y  =  ^h. 
For,  if  possible,  let    x  =  a-^c,     and  c  be  rational; 
then  \'  a-{-c  +  \^y  =  a+  4/J,  [byp- 

.*.  V^  —  Vy~  ^y     which  is  impossible,  [th.  7. 

,\x  =  a,    ^y—^h. 
Cor.  2.7/    x-\- /^y  =  a-{- \^h,       then      x—i^y  —  a—  \/h. 
Cor.  3.  If  i^x-\-  j^y^  j^(a  +  \/h),  then  \/x  —\^y  —  \/(a  —  |/Z»). 
For  \'  x  +  y^-2i^xy  =  a-\-\^h,  [sqr.  both  mem. 

,\x-\-y  —  a    and    2  j\/xy  =  ^b,  [cr.  1. 

,\x  +  y  —  2^xy  =  a  —  ^h, 
and    t^x  —  |/y  =  |/(«  —  V^).  Q.E.  D. 


PR.  i4,  §6]  ROOTS   OF   BINOMIAL   SURDS.  193 

PrOB-  14.    To    FIXD  A  SQUARE    ROOT  OF  A    BINOMIAL  SURD. 

Let  «+  4/^  be  a  binomial  surd  ;iik1  let  ^x  +  \/y  =  \/{a  4-  i^b); 
then  \'  |/.r  -Vl/=  i'(/^  -  V^)>  [^^-  ''  ^r.  3. 

QUESTIONS. 

I.  In  the  proof  of  theor.  6,  |/a,,  |/b,  being  surds,  |/(b/a) 
can  be  rational  only  when     b/a  =  1!  state  the  proof. 

2o  The  equation  ai/x  =  b\/B  is  satisfied  when  a= —b, 
a  =  b;  but  not  when  a  =  ^,  a=  -b,  nor  when  a=  —b,  A=  —  B. 

3.  If     a-^b^c  =  a/  +  byc\     then     a  =  (f/,      b-b\    c-c\ 

4.  If     i^a  -{-^b  =  '[/c  +  \/d,     the  surds  are  equal  in  pairs. 

5.  A  quadratic  surd  cannot  equal  the  sum  of  two  other 
non-conformable  quadratic  surds,  nor  their  difference. 

6.  If  5  -2|/6,  2|/6  -5  be  the  two  square  roots  of  49-20^/6, 
and  |/3-V2,  \/2-\/-6,  (^3- >|/2)|/-1,  (4/2- 1/3)|/-1  be 
its  four  fourth  roots,  what  is  the  relation  between  the  two 
square  roots?  between  the  four  fourth  roots  ? 

7.  If   4/5  +  1/ -3    be  one  of  the  fourth  roots  of  a  binomial 
surd,  what  are  the  others  ?  what  are  the  square  roots  ? 
Find  a  square  root  of: 

8.  7  +  2|/10.  9.  7  +  4|/3.  10.  2-4/3. 

II.  16-6|/r.  12.  4/IS-4/I6.  13.  9  +  24/I4. 
14.  8|/3-6|/5.       15.  75-12|/21.            16.  V-'  +  ^l^- 

17.   -12  +  64/3.      18.  24/[l  +  (l-r)^].     19.  \-2a^(l-a^), 
20.   ab-{-e  +  V{b''-c'){a--c').       21.  xij-2x{xij -x'Y^' 
22.  9+24/3+24/5  +  24/15.  23.  2(3-4/2-4/3  +  4/6). 

Find  a  fourth  root  of: 

24.  28-I64/3.       25.  49  +  2O4/6.  26.  137-364/14. 

27.  (f}  +  b~  +  Qab-^{a^/W-^a''W^).  28.  -8  +  84/-3. 


194  POWERS   AND   ROOTS.  iVI,  Prs. 

Note  L  It  a^—bhe  a,  perfect  square,  so  are  a  +  \/b,  a  —  \/b. 

For  '/  \^x  +  \/y  =  \/{a  +  \^b),     and     ^x-\/i/  =  \/(a  -  \/b), 

.\x-y-\^{a^-b), 
and     \/{d^-b)     is  rational  if  the  root  sought  be  possible. 
E,g.,  7  +  4|/3     is  a  perfect  square; 

for     a  — If     J  =  48,     a^  — J  =  49  — 48  =  1,     a  perfect  square. 
But  7  +  2|/3     is  not  a  perfect  square; 

for  rt  =  7,  b  — 12,  a^  -  J  =  49  — 12  =  37,  not  a  perfect  square 
Note  2.  Since  (|/a;  +  4/?/)^  =  ^  +  ?/ +  2 1/^:^,  it  appears  that 
the  square  root  of  a-{-2\/b  is  the  sum  of  the  square  roots 
of  two  numbers  whose  sum  is  a  and  whose  product  is  b.  This 
consideration  often  makes  it  possible  to  find  such  roots  by 
inspection. 

Kg.,     |/(-4  +  4/-84)  =  |/(-4  +  2|/-21); 
and  V  -7  +  3= -4,      -7x3= -21, 

.\4/(~4  +  V-84)  =  y-7+|/3. 
So,     4/tt  +  4/5)  =  t^(|  +  24/f)  =  l  +  i4/5. 
So,     |/(|/32  -  ^24.)  =  V(4|/2  -2^6)  =  i/(3|/2)  -  |/(|/2) 
=  ^18-1^2. 

PkOB.   15.    To    FIND   A   CUBE    ROOT   OF   A   BINOMIAL   SURD. 

Let    a  +  \/b  be  a  binomial  sui-d,  and  x-\- \/ij  =  \/{(i-\- ^b)\  (\) 
then  *.'  0?  +  2>xy  +  (3:?;^  +  y) \/y  =  ai-  \/by    (2)     [cube  both  mem. 

.•.a:3  +  3:r//-(3.?;2  +  ?/)4///  =  rt-i/^    (3)  [th.  7  cr.  2. 

and       x  —  \/y-\^{a  —  \/b),  (4) 

.".  x^  —  y—  ^(a^  —  b),  [mult.  eq.  1  by  oq.  4. 

and  the  root  is  possible  only  if     \/{ar~b)     be  rational. 
Pu  t  m  for     ^{a^  -  b) ;    then     y  =  x^  —  m, 
and  •."ar'  +  3a:?/  =  «,  [add  eqs.  2,3. 

, %  4.r  —  3 mx  =  a.  [el i m.  y. 

From  this  point  on  there  is  no  general  solution,  but  partic- 
ular examples  may  be  solved  by  finding  a  value  of  x,  by  in- 
spection, from  the  equation     ix^  —  3?iix  —  a, 


14,15.16]  ROOTS   OF  BINOMIAL   SUKDS  195 

E.g.,  to  find  a  cube  root  of     10 4- 64/8; 

then  %•  a  =  10,     l^  108,     m  =  f  (100  - 108)  =  -  2 ; 

.-.  4:fH6.i'  =  10, 

.'.  a;  =  l,     ?/  =  3;     and     l4-|/3     is  the  root  sought. 

QUESTIONS. 

1.  Any  coefficient  may  be  prefixed  to  a  radical,  if  the  num- 
ber under  the  radical  sign  be  divided  by  that  power  of  the 
coefficient  whose  exponent  is  the  index  of  the  radical. 

By  inspection  find  a  square  root  of : 

2.  8-2j/7.  3.  3|/5  +  |/40.  4.  2i-|/2. 

5.  x^  +  x  +  2x\^x.        6.   -5-2|/-6.       7.  2a-2x/(d^-h% 
8.  2«-&-2|/(rt2-«Z>).  9.  2rt  +  ^  +  2|/(r^2_j.^^^j^ 

Find  a  fourth  root  of : 

10.  17  +  12-^/2.  11.  49-20|/6.  12.  14  +  8|/3. 

13.  89  +  28|/10.        14.  -221-60|/-1.  15.  4^*. 
Find  a  cube  root  of : 

16.  7  +  5|/2.  17.  16  +  8|/5.  18.  45-29|/2. 

19.  22  +  10|/7.  20.  38  +  17^/5.  21.  2I4/6 -284/5. 

22.  3rt-2«H(l+2^^2)|/(l-«').  23.  1 +  3«  + (3  4-f0i/^. 

Of  the  binomial  surds  below,  which  are  perfect  squares  ? 
24.  4i-V5i    25.  3-4/2.    26.  V^y-V{^'/y)-   27.   9-3|/'5. 

28.  If  \/x  + 1/1/  +  4/2;  =:  t/(«  +  2|/^  +  2 |/c  +  2 4/fZ),   2:,  ij,  z  m ust 
satisfy  the  four  conditions 

x-^y-\-z  —  a,    xy  —  l),    xz  —  c,     yz  —  d. 
Find  a  square  root  of     6  4- 24/2  +  24/3  +  24/6. 

29.  The  square   root   of     10  +  24/6  +  24/14  +  2-^/21     cannot 
be  expressed  in  the  form     \/x  +  4/^  +  4/^. 

Find  a  square  root  of : 

30.  10  +  24/6  +  24/10  +  24.15.  31.  8  +  21/2+24/5  +  24/10. 
32.  15-24/15-2^21+24/35.  33.  11+2^6+44/3  +  6-^2. 
34.  15-2^/3-24/15  +  6^2-24/6  +  24/5-24/30. 


196  POWERS   AND   ROOTS.  [VI. 

§7.    QUESTIONS   FOR   REVIEW. 
Define  and  illustrate: 

1.  A  power;  a  root;  a  base;  an  exponent;  a  root  index. 

2.  A  fraction  power;  a  commensurable  power. 

3.  Powers  of  a  base  in  the  same  series;  like  powers. 

4.  A  radical;  a  radical  factor;  a  radical  expression. 

5.  A  rational  radical;  a  real  radical;  a  surd;  an  inl!lginar^^ 
C.  A  simple  radical;  a  quadratic,  a  cubic,  a  quartic,  and 

a  quintic  radical. 

7.  Like  radicals;  conformable  radicals. 

8.  A  binomial  surd;  a  trinomial  surd. 

9.  A  pair  of  conjugate  quadratic  surds;  a  pair  of  comple- 
mentary surds;  a  group  of  complementary  surds. 

State  and  prove: 

10.  The  binomial  theorem. 

11.  The  principle  by  which  a  commensurable  power  of  a 
commensurable  power  is  found. 

12.  The  principle  of  equal  fraction  powers. 

13.  Tlie  principle  by  which  the  product  of  two  commen- 
surable powers  of  the  same  base  is  found. 

How  does  this  principle  apply  in  finding  their  quotient  ? 

14.  The  principle  by  wliicli  the  product  of  like  commen- 
surable powers  of  different  bases  is  found. 

How  does  this  principle  apply  in  finding  their  quotient? 

15.  The  principle  of  the  equality  of  tlie  like  parts  of  two 
equal  simple  surds. 

IG.  Tlie  principle  of  the  inequality  of  two  non-conformable 
simple  surds. 

17.  The  principle  of  the  equality  of  the  like  parts  of  two 
equal  simple  bincmiial  quadratic  surds. 

18.  What  is  the  product  of  two  conformable  simple  quad- 
ratic surds?  their  quotient?  the  product  of  two  such  non- 
conformable  surds?  their  quotient? 


$7]  QUESTIONS   FOR   REVIEW.  197 

Give  the  general  rule,  with  reasons  and  illustrations,  for: 

19.  Reducing  a  simple  radical  to  its  simplest  form. 

20.  Freeing  a  simple  radical  from  coefficients. 

21.  Adding  and  subtracting  radicals. 

22.  Multiplying  and  dividing  radicals. 

23.  Getting  powers  and  roots  of  radicals. 

24.  Reducing  a  fraction  with  a  surd  denominator  to  an 
equivalent  fraction  with  a  rational  denominator  when  the  surd 
denominator  is  a  monomial;  when  it  is  a  simple  quadratic 
surd;  when  it  is  a  binomial  quadratic  surd  containing  a  com- 
plex radical;  when  it  is  any  binomial  surd. 

25.  Rationalizing  an  equation  that  contains  surds. 

26.  Finding  the  square  root  of  a  polynomial. 

27.  Finding  the  cube  root  of  a  polynomial. 

28.  Finding  the  root  of  a  polynomial  when  the  root  index 
is  a  composite  number. 

29.  Finding  the  ?ith  root  of  a  polynomial  when  7i  is  prime. 

30.  Finding  the  square  root  of  an  integer,  and  of  a  decimal 
fraction. 

Explain  the  principle  of  dividing  the  numeral  into  periods; 
and  that  of  contraction. 

31.  Finding  the  cube  root  of  an  integer,  and  of  a  decimal 
fraction. 

Explain  the  principle  of  dividing  the  numeral  into  periods; 
and  that  of  contraction. 

32.  Finding  a  root  of  a  fraction. 

33.  Finding  a  square  root  of  a  binomial  quadratic  surd. 

34.  Finding  a  cube  root  of  a  binomial  quadratic  surd. 

35.  If  x^  +  62;^  +  '7x^  —  6x-\-  m  be  a  perfect  square,  what  is  7n  ? 

36.  If  4:X^-\-12x^-\-5x^  —  23^-h7nx^-{-nx+2J  be  a  perfect 
square,  what  are  the  values  of  ?/?,  n,  p? 

37.  Apply  the  square  root  process  to  factoi-ing 

^x'  +  12x1/  +  8^'  +  IQxz  +  22i/z  +  15z\ 


198  QUADRATIC  EQUATIONS.  [VU,i>Ba. 

Vn.  QUADRATIC  EQUATIONS. 


An  equation  that  is  of  the  second  degree  as  to  its  unknown 
elements  is  a  quadratic  equation. 

E.g.,    a;^  =  9,     :2^-f3.?:=l8,    ax^'^bx  +  c-O,  [z  uukn. 

So,       xy  ■=  12,    ax^  +  2hxy  -i-  by^  +  2gx  +  2fy  +  c  =  0,  [x,  ij,  unkn. 
So,      ax^  +  hy^  +  cz^  +  2fyz  4-  2gzx  +  2A.r?/  =  0.        [x,  ^,  z,  u n  kn. 

§1.   ONE   UNKNOWN  ELEMENT. 

An  equation  of  the  form    a;^=:9,    is  an  i7icomplete,  or  pure^ 
quadratic  equation;  one  of  the  form   x^  -{-^x—  18   is  complete. 

PrOB.  1.    To  SOLVE  AN"  INCOMPLETE  QUADRATIC  EQUATION. 

Reduce  the  equation  to  the  type-form    x^  =  q,    and  take  the 
square  root  of  each  member;  then  a:=  ±  /^q,   [III,  ax.  7. 

Kg.,  if  i(a;2-10)+TV(C''^''-100)  =  32;2-65: 

then     10a;2 - 100  + 182)2 _  300 -90.^;2- 1950,  [nuilt.  by  30. 

-62a;2=-1550,    x^=:2b     and     a:=±5. 
There  are  two  square  roots,  opposites;  they  are  botli  real  if 

the  absolute  term  be  positive,  and  both  imaginary  if  it  be 

negative.     It  may  seem   that   the  last   equation   sliould   be 

±2'=  ±5;    but  this  gives  no  new  roots. 

PrOB.  2.    To   SOLVE   A   COMPLETE   QUADRATIC   EQUATION. 

Reduce  the  equation  to  the  type-fonny    x^+px  +  q  —  O', 
trans2)0se  the  absolute  term,  and  to  each  member  of  the  equation 
add  the  square  of  half  the  coefficient  of  the  fird-deifree 
term;  [II,  pr.  3  nt.  4  frm.  3,  4. 

take  the  squai^e  roots  of  these  sums,  and  solve  the  two  simple 
equations  so  found,  [111,  ax.  T. 

The  result  is  of  the  form    x=  -ip±i\/(p^~4q). 
E.g,,if  if2  +  32;rz40: 

then       a^-\-3x  f  21-^42i,  (add  (3/2)*^  to  earh  tnom. 

(g+l^-  jzGi,  [take  sqr.  rts, 

,^=-1.1  ±0.1:^5     or     -8. 


i.2.§ii  ONE  UNKNOWN  ELEMENT  199 

QUESTIOXS. 

1.  Make  a  quadratic  equation  to  state  that:  the  area  of  a 
square  13  4225  square  yards;  the  area  of  a  rectangle  is  1200 
square  rods;  the  sum  of  the  squares  of  two  numbers  is  three 
times  their  product;  the  product  of  the  sum  and  difference  ol" 
two  numbers  is  33;  the  product  of  two  numbers,  one  5  less 
than  the  other,  is  24;  the  sum  of  the  squares  of  tliree  numbers 
increased  by  twice  tlieir  products,  two  by  two,  is  36. 

Which  of  these  equations  are  complete? 

2.  How  can  two  independent  simple  equations  be  obtained 
from  one  quadratic  equation  ?     Write  the  forms  for  the  two 
roots  separately,  and  find  their  sum  and  their  product. 
Solve  the  pure  quadratic  equations: 

3.  (x'+i)(.i''i-2)  =  {a^^Q)(:o^-l). 

5.  i(3:6-2-7)  +  l(25-4:X^)  =  ^ox'-U). 

6.  3(5.T«-7)(35-22■)+27(5.^'2-|-7)  =  9(5.^•2-7)(17-|2;). 

7.  Why  is  an  incomplete  quadratic  equation  called  a  pyre 
quadratic  ?  a  complete  equation,  an  affected  quadratic  ? 
Solve  the  complete  quadratic  equations: 

S.a?-bx  +  Q  =  0,      9.  a;2-82:  +  15  =  0.    10.  rc^ 4- 102;= -24 

11.  0^-52^  +  4  =  0.    12.  62;^- 192; +  10  =  0. 

13.  72.-2- 32:  =1G0.    14^  1102.'2-2Lt  +  1  =  0. 

15.   (2;-2)-^-2(.r  +  2)-i  =  3/5.  16.  |/(2.2:  +  5)=a;  +  l. 

17    32;-2     22;-5_8  x  +  a     x  +  b     x  +  c _ 

'  2x-6     ^x-2~3'        ^^'  x-(ix-b'^x-c~ 

xj^     x-3^2x-3  ^Z^  .  _^__7 

•  2;  +  2^2;-2      2;-l*  2x^x  +  d~\0' 

X  +  3     x  —  4:         2  (i  +  b-\  X     a      b     x 

23       ^^  b    _    2c  ^^    a+cja+x)      a+x  a 


a     x  —  b     x  —  c  '  a  +  c{u  —  x)         x        a  —  2cx 


200  gOADRATlC  EQUATIONS  IVU.Pr 

SPECIAL  CASES. 

Note    1.   The  roots  of  the  equation     ct^  h px  •]- q  =  0    are 

-iP^l  A  /^'  -  ^^)^     -  il^  -  I  V(P'  -  '^9)>  [above, 

whose  values  depend  upon  the  values  of  p^  q. 

There  are  four  special  cases : 
{a)  p posiiive,  q  negative: 

two  real  root-,  the  larger  negative,  the  smaller  positive. 
{h)  p,  q  loth  n:gative : 

two  real  roots,  the  larger  positive,  the  smaller  negative. 

(c)  /?,  q  loth  positive : 

two  real  roots,  both  negative,  ix    p^  —  \q     be  positive; 
two  real  roots,  both  negative  and  equal  to  -\]),  if  p^-^q  —  ^\ 
two  imaginary  roots,  conjugates,  if    p^  —  Aq     be  negative. 

(d)  p  negative,  q  positive : 

two  real  roots,  both  positive,  if    p^  -^q     be  positive; 

two  real  roots,  both  positive  and  equal  to  ~\p,  if  p^-\q  —  ^\ 

two  imaginary  roots,  conjugates,  if    p^  —  ^q     be  negative. 

THE   SUM   AND   THE   PRODUCT   OF  THE   ROOTS. 

Note  2.  The  sum  of  the  two  roots  is  -p,  and  their  product 
is  q.    A  quadratic  equation  can  have  not  more  than  two  roots. 

THE   ABSOLUTE   TERM  ZERO. 

Note  3.  If    <7  =  0,     then  of  the  equation    o^Vvx  —  ^S    the 
two  roots  are    0,  -p,    both  real. 

SOLUTION    BY    FACTORING. 

Note  4.  If  the  ex[)ression     x^  f-px-j-q     be  readily  factored, 
then  each  factor  maybe  put  equal  to  zero,  and  the  two  simple 
equations  so  found  give  two  values  for  x. 
K.g.,  to  solve  the  equation     x^-bx^6  =  0\ 
then  va;^-5a:f6  =  (r-2)(z-3), 

and       this  product  is  0  whether     ^-2  =  0     or    a; -3-0, 
.*.  the  roots  are  2,  3. 

The  equation  is  found  by  subtracting  the  roots  in  turn  from 
X  and  equating  the  product  of  the  remainders  to  0. 
fcl.g.,  if  the  roots  be  2,  3,  the  equation  is     (^  -2)(a;-3)=0. 


2.§il  ONE   UNKNOWN   ELEMENT  201 

QUESTIONS. 

1.  Wbatevor  be  the  sign  of  p,  what  is  that  of;/? 

2.  If  q  be  negative,  wliat  is  the  sign  of    jr  —  iq  ? 
Wbat  does  this  show  about  the  character  of  the  roots  ? 
Is     iViP^  —  '^Q)     t^^en  larger  or  smaller  than  !/>  ? 

3.  If  p  be  positive  and  q  negative,  wliich  is  the  larger. 

-iP  +  W{p'-^9)     or     -y-l^{j/-4q)? 

4.  If  q  be  positive  and    7^  =  45',     what  are  the  roots  ? 

5.  If    ;/>45',     are  the  roots  real  ?  if    ;/<4^? 

6.  If  q  be  0,  what  is  the  product  of  the  two  roots  ? 

7.  If  p  be  0,  of  what  kind  is  the  quadratic  ? 

Write  the  sum  and  the  product  of  the  roots  of  the  equation: 

8.  x^-4:X  =  G0.         9.  3x-2 +  6.7-24.         10.  5ci^-lbx  =  U0. 
11.  2:6"2-3.r  =  12.      12.  x^-ax^O.  13.  5:^^- 15a;  =  200. 

14.  Suppose  the  quadratic  equation  af-{-px  +  q  =  0  to  have 
three  different  roots,  r,  r\  r"  \  what  is  the  value  of  ?v' ?  of 
rr"  ?   Prove  that  two  of  the  supposed  three  roots  are  the  same. 

15.  If  ?',  r'  be  the  roots  of  the  equation  Q?—px-\-q  =  {), 
find  the  value  of     r/r'  +  r'/r;     of     r^-f-r'^;     of     r^-\-r'\ 

16.  Write  the  expression  x^  —  {a  +  b)x-\-ab  in  the  form 
{x~a){x  —  I)):  what  is  the  value  of  this  product  if  x  —  a  =  0? 
if    x  —  b  =  0?    What  are  the  roots  of  the  equation  ? 

Form  a  quadratic  equation  whose  roots  shall  be: 

IT.  4,    -5.      18.  2|,  2.      19.   -f,    -8.      20.  a-{-b,  a-b. 

Bv  factoring,  solve  the  equations: 

21.  ?/H13?/  =  14.       22.a;2  +  .r^^3Q^      23.4.-^  +  12.^  +  9  =  0 
24.  2r*-«*=0.  25.a^-5x  =  U.      26.  7^ -^-ci^-x-l-O. 

27.  9a:2-30.r  +  25  =  0.  28.  x^  +  ex^-4:X=r.2i. 

Put  the  functions  below  equal  to  0,  solve  the  equatiors  so 
formed,  and  by  aid  of  their  roots  factor  the  functions: 

29.  ex'-ldx-tl^.    30.  (x-af-b^    31.  a^-'2mx-\-m'-7i\ 
32.  x^-{77i-{-7i)x-{-{m+p){7i-p).         33.  .T^  +  ^r  +  l. 
34.  7^-ax-2a^-b^  +  3ab.  35.  x^-x  +  l. 


202  QUaMATIC  equations.  IViI.Pr. 

GENERAL   FORMS. 

Note  5.  The  rule  for  solving  complete  quadratic  equations 
may  be  stated  in  a  more  general  form: 
Reduce  the  equation  to  the  ^^jje-/or^/i     ax^-hbx  +  c  =  0; 
multiply  the  equation  by  any  7iumher  that  makes  the  coefficient 

of  the  first  term  a  perfect  square; 
make  the  first  member  a  jierfect  square,  take  the  square  root 
of  each  member,  and  solve  the  simple  equations  so  found. 

With  a  proper  multiplier  fractions  may  be  avoided. 
E.g.,  if  h  be  even,  multiply  by  a-,  if  odd,  by  4rt.      , 

Both  rules  rest  on  the  form  assumed  by  the  square  of  a 
binomial,  as  does  that  for  finding  the  square  root. 

The  roots  are  [-b-}- ^(b^-4:ac)]/2a,    [-b- ^(i;^-4ac)]/2a. 

There  are  three  special  cases: 

(a)  c  zero:  then    x  =  0,    x= —b/a,    two  real  roots. 

(b)  b  zero:  then     x=  ±:\/{  —  c/a), 

two  real  roots,  opposites,  if  a,  c  be  of  contrary  signs; 

two  imaginary  roots,  conjugates,  if  a,  c  be  of  the  same  sign. 
{c)  a  zero :  i\iQ\\     x={-b-\-b)/0,     x  =  {-b-b)/0, 
i.e.,    2*  =  0/0,    .'c  =  Qo,    an  indeterminate  and  an  infinite  root. 

But  this  indeterminate  root  can  be  determined. 
For  if    a  =  0,    and    x^oo ,    the  equation  becomes    bx  +  c  =  0, 
whose  single  root  is     —c/b. 

Tliis  case  is  specially  important  as  showiiig  the  values  of 
X  if  a  be  thought  of  as  taking  changing  values  and  growing 
smaller  and  smaller;  for,  tiien,  as  ((  grows  very  small,  one  of 
the  roots  grows  very  large,  and  the  other  approaches     —c/b. 

This  is  also  evident  if  the  equation  be  written  in  the  form 
x~\b-\-cx-^)=  —a,  [ (1  i V.  eq.  bx -\-c=  - ax^  by  x\ 

For,  if  a  grow  very  small, 

then  either     x~'^     grows  veiy  small  and  x  very  large, 
or     b  +  cx~^     grows  very  small  and  ./•  a})pr()a('lK'S     —c/b', 
and  both     oo     and     —c/b     satisfy  the  equation  wlien     a  =  0. 


^\§l]  ONE   UNKNOWN   ELEMENT.  203 

QUESTIONS. 

Solve  the  equations: 

1.  0.-2  + ix— -10/9.     2.   6x^  +  9:c  =  Sl.    3.  8a;2-2l2:  +  f  =  0. 
4.  x^  -  2mx  +  m^  =  n\  5.  Q^x^  - 1  Ix  =  10.   6.  I  x^  _  i^^;  +  30  =  0. 
7.  «V  +  2rt5:6-  +  Z»2-c2  =  0.'  8.   «Z'2:H(3^-4«).7;  =  12. 

9.  Find  the  sum  and  tlie  product  of  the  roots  of  ax^  +  tx  +  <" 
If     h  —  ^,     of  what  kind  is  the  quadratic?  what  is  the  sum 
of  the  roots?  wliat  relation  to  each  other  have  they? 

If   c/a   be  negative,  what  relation  have  the  roots?  if  c  —  a'^. 

10.  If  in    a:(?-\'hx-\-c,    «  =  0,    what  is  one  value  of  .r  ? 
Show  that  the  product  of  the  roots  is  then  infinite,  and 

hence  find  the  other  root. 

11.  Discuss  the  equation  ax^-^-hx^-c  —  ^,  after  the  manner 
of  note  1,  and  show  that  the  two  roots  are:  real  and  unequal, 
if   lr>\ac\  real  and  equal,  if  ^^  =  4ac;   imaginary,  if  b'^<^ac. 

Of  the  real  and  unequal  roots  which  is  the  larger?  What 
conditions  make  the  real  roots  both  positive?  both  negative  ? 

12.  Jf  ?•,  r'  be  the  two  roots  of  ax^^-tx^-c,  then  ax?-\-'bx-\-c 
may  be  factored  and  written  in  the  form     a{x—]')(x  —  r'). 

13.  What  form  has  a  quadratic  equation  whose  roots  are 
opposites  ?  reciprocals  ? 

14.  For  what  value  of  c  will  the  equation  2:<^  +  6:6' +  6' =  0 
have  equal  roots?  reciprocal  roots  ? 

15.  If  r,  r' be  the  roots  of  the  equation   .-c^+;;.r  +  ^  =  0,   find 
the  equation  whose  roots  are  —  ?•,  —  r';     1/r,  \/r'. 
Without  solving,  show  that  the  equation 

1 6.  x^ ±  2(/?  +  q)x  +  2(7/  +  ^2j  =  0     has  imaginary  roots. 
IT.  a;^±2(7J  +  ^)a:  +  (;^  +  ^)-  =  0     has  equal  roots.  , 

P'or  what  value  of  m  will  the  equation 

18.  a;^-15-??i(2a;-8)  =  0     have  equal  roots  ? 

19.  {x^ -  bx)/{ax  -c)-  {m  -l)/{m  + 1),     opposite  roots  ? 
Without  solving,  tell  the  signs  of  the  roots  of  the  equations: 

20.  x^-bx  =  3G.         21.  a-2  +  5:/:  =  14.  22.  x''  +  j%x=l/5. 


204  QUADRATIC  EQUATIONS.  :vu,Pr. 

EQUATIONS   SOLVED   AS   QUADRATICS. 

Note   6.    Equations   of    the  form     <(:t^"  -\-bx^-{-c  =  0,      or 
(«a;^"  +  bx""  +  cf^  -\-p(ax^''  +  ix""  +  c)"*  4-  ^  =  0,    are  solved  as  quad- 
ratic equations. 
E.g.,  if     9x'-52x^  +  G4:  =  0: 

then  •.•  81a;*-468a;2  +  67G  =  100,  fmult.  by  9,  add  100. 

.-.  9^:^-26  =  ±  10,  [ti^ke  sqr.  rts. 

.•.ic2  =  4or  16/9, 

.*.  a;=  ±2  or  ±4/3,    four  real  roots. 
So,  if     (92;*-52a:H80)H9(9.'C*-52a:2  +  80)-400  =0: 
thenv4(9:c*-52:z;^  +  80)H36(92;*-52^-2  +  80)+81  =1681, 
.•.2(9a:*-522;2  +  80)=-9±41  =  32  or  -50, 
.-.  9a;*-52a;2  +  80=16  or  -25, 
and       x=±2,     ±4/3,      ±i|/(2G±  4/-269),     eight  roots. 
So,  if    x'  -  Qx"  +  4:x^  +  16x  =  14 : 
then      {x^  -  3xf  -  5  (rc^  -  3x)  =  14, 

a.-«-32:  =  5/2±|/(81/4),  [solve  for  t'-^x. 

x^-dx  =  '7  ov  -2, 

ic  =  3/2  ±  i|/37     or     3/2  ±  1/2,  [solve  for  x. 

x=l{3±:\/37),  2,  or  1,  four  roots. 
So,  if    3.T2-2|/(3xH2a;-7)  =  10-2a:: 
then      3x'-i-2x-7-2.s/(3x'-h2x-'7)  =  3, 
|/(3a;H2a;-7)  =  l±2  =  3or  -1, 
^x^  +  2x-7^9  or  1, 
ic2  +  |a;=16/3or8/3, 
a;z::-l/3±7/3     or     -l/3±:5/3, 
x=2,      -8/3,     or     4/3,     -2,     four  roots. 


2.  §n  ONE   UNKNOWN    ELEMENT.  205 

QUESTIONS. 

Solve  the  equations: 

1.   i/(2x-+:)  +  |/(3.r-lS)  =  |/(r.t'  +  l).     2.  3.7;  +  2|/.7;-l  =  0. 

3.  2-^  +  3  =  2^(./--2.7;  +  2)+2.^'.  4.  a;^/" - 1 3./ ' '^'^  =  1 4. 

5.   ^(x^-2x  +  9)-ix^  =  3-x.  6.  x*-U.r -h 4.0  =  0, 

7,  3.t2  +  15.?'-2|/(:?;2  +  5.c  +  l)  =  2.        8.  a;^'Vtc-'''=3'/*. 

9.   (:r  +  x-6y-4:(:L^-\-x-G)  =  12.  10.  >/2.T-7.rrT -52. 

11.   {af  +  xY-^x^-{-x)  =  10S.  12.  wa;2  +  a;  +  ;i  +  l=:0. 

13.  :/;  +  5-|/(.r  +  5)=:6.  14.    f'i^^  +  3i^x  =  18. 

15.  2.r2  +  6  +  3|/(2;/r  +  6)^10=  16.  x\lQ -\-x^)  =  2l6, 

IT.  (.i-2  +  2)-  +  198  =  29(./;=^  +  2).  18.  4  =  5x-2-:6-*. 

19.   j/(a:  +  4)-|/.r=|/(.c  +  |).  20.  3./;«  +  8x*-S.f2^3. 

21.  {x-^l/xY-4:{x  +  l/x)  =  2i,  22.  ^-^.^^^.^^.^^q^ 

23.  2(f.'r2-|)2  +  5(fa:2-f)=i63.  24.  af  +  a^^/x^  =  a'  +  bl 
25.  5|/(3/.r)+74/(.r/3)  =  22|.         26.  x'  +  2x^-3x'-4:X=d6. 

27.  ^HlA'  +  2(:6'  +  l/a;)  =  9i.  28.  SS/.^^- 10^  =  3. 

(2:-5)(a;-c)      (^' -  «)  (.i;  -  c)      (.T-f^)(.7;-<5') 

30.  7/[i/(.r-6)+4]  +  12/[^07;-6)+9]+l/[4/G7;-6)-4] 

+  6/[V(:^'-6)-9]  =  0. 

31.  m%x  +  m  +  17?i){x-7n  +  7nY 

=  n^  {x  + 177)1  +  71)  (x  +  7w  -  7i)K 

33.  [x -  |/(x^ -  a')y^[x  +  i/('^-' - «')] 

=  {/{j^ - a')[^(,Tf  +  ax)  -  ^{x' - ax)l 

34.  x''-3x  +  4:-{-2i/{:yy'-3x-i-6)  =  6. 

36.  (rta;2"  +  bx""  +  c)^'"  ±  (dx^"  +  f .i"")""  =  0. 


206  QUADRATIC  EQUATIONS.  \vil,Pu. 

§2.    TWO   UNKNOWN   ELEMENTS. 

PrOB.  3.  To  SOLVE  A  PAIR  OF  EQUATIONS  INVOLVING  THE 
SAME  TWO  UNKNOWN  ELEMENTS,  ONE  EQUATION  SIMPLE,  THE 
OTHER   QUADRATIC. 

Eliminate  one  of  the  unhnown  elements  from  the  quadratic 

equation;  [IIIj  pr.  2. 

solve  the  resultajit  for  the  other  unhnoivn  element  and  replace 

this  element  hy  its  value  in  the  simple  equation; 
solve  this  equation  for  the  first  unhnoicn  element. 
E.g.,  if  3a; +  2^  =  20,     3:«H5a;^  + 7^^  =  425: 
then  •.•  IT  =  -^(20  — 2?/),  [sol.  first  eq.  for  a\ 

...  1(20 -2?/)H  1^(20 -2?/) +7^2  =  425,  [repl.  x  in  sec.  eq. 
/.  15/  +  20?/  =  875,     ?/  =  7  or  -  SJ,  [sol.  quad,  for  y. 

.'.  3^;  +  2  •  7  =  20,     x  =  2,  [repl.  y  in  first  eq. 

and       3a; -2- 8^=20,    a;  =  12f, 
and  the  two  pairs  of  roots  are     2,  7;     12f,  —  8|. 

Check.    Both  pairs  of  roots  satisfy  the  quadratic  equation. 
If  the  two  equations  be  such  that  they  can  be  combined  in 
one  of  tlie  familiar  forms     x^±%xy-{-y'^,     x^  —  y^,    the  work  is 
shortened  by  the  use  of  such  form. 
E.g.,  if  a;  +  ?/  =  13,    xy-Vl\ 

then  *.•  x^  +  Ixy  +  ?/^  =  169,  [sqr.  first  eq . 

and       4.«/  =  48,  [mult.  sec.  eq.  by  4. 

.*.  x^  —  2xy  +  7/*^  =  12 1,  a;  —  y  =  ±  11 ,    [sub.  and  get  sqr.  rts. 
.*.  from  the  equations     x-\-y—V'S,     x  —  y—W,         come 
a;  =  12,     ?/  =  !; 
and       from  the  equations     x-^y—Vc*,     x  —  y——\\,     come 
x^\,     ?/  =  12. 

So,  if    x^-\-y^  =  4:6,     x  —  y——^: 

then  '.•  x^  —  '^xy-\-if  =  Sl,  [sqr.  sec.  oq. 

.-.  ^xy=  -36,    a;2  +  2a;.?/4-.?/  =  9,     x^-y=  ±3, 

,\x=—Q,    y=^;    x=-3,    y-Q. 


3,§^M  TWO   UNKNOWN  ELEMENTS.  207 

QUESTIOXS. 

Find  the  values  of  x,  y  from  the  pair  of  equations: 

1.  x  +  ij  =  l,    a;2  +  2/=34.  2.  x-y  =  l2,    x^  +  y^=74, 

3.  x-\-y  =  a,    xy  =  b^.  4.  x  —  y  —  a,     xy  =  b\ 

5.  3.?;-5«/  =  2,     xy  =  l,  6.  iC  +  ?/=100,     a;?/ =  2400. 

7.  x  +  y  =  a,     x'^f^zl^,        8.  a;i/2  +  ?/- 1/^  =  4,     rc-2/-i  =  8. 
9.  a;  +  ?/  =  4,  2;-i  +  ^-^  =  l.     10.  2a;  +  3?/  =  37,  a;-i  +  «/-^  =  |t- 
11.  a;  +  ?/  =  2,  a;2-2a;.y-.v^=l.    12.  a;  +  ^  =  18,  x^^f^^^W, 
13.  a;  +  «/  =  72,    -^^+|^i/  =  6.      14.  x-y  =  \%,  a;3-?/^  =  4914. 

15.  a;2^-i  +  i/'^-'  =  9,  a;-^  +  ?/-i  =  3/4. 

16.  In  the  pair  of  equations    x-\-y  —  \Z,     xy=12,    how  are 
the  results  affected  if  x,  y  exchange  places  ? 

Show  why  either  x  or  //  may  be  12,  and  the  other  be  1. 

17.  In  the  pair  of  equations    :/;^  +  ^^  =  45,    x  —  y=—d,    can 
X,  y  exchange  places  ?  x,  —y?    What  relation  have  x,  y? 
Solve  the  pair  of  equations : 

18.  x-y  =  b,     xy  =  126,        19.  x-\-y  =  8,     x^-t/=ie. 
20.  x-y  =  4:,     a;2-?/2=32.     21.  ^  +  ^  =  11,     x^  +  y^  =  407. 
22.  x-y  =  4:,x^-y^  =  98S.     23.  3a; - 4?/ =  4, 9a;^- 16^/2  =176. 
24.  x^-y''  =  21,     x{x  +  yY  =  4:0.     25.  x-y  =  2,c(f-f=992. 

26.  3x-2y  =  lO,     3.r-ii7/-/=80. 

27.  x/y-y/x  =  3/2,    x~y^l.      28.  a;  +  ?/  =  2,  af +  ?/^=992. 

29.  l/a;Hl/i/'=  126/125,     l/x  +  \/y  =  Q/f). 

30.  a;  +  |.y=ll,     rc3  +  2.^2!/  +  f.2:/  +  ^?/^  =  1331. 

31.  5.r-?/=3,     ?/2-6.^'2=25.  32.  x-y-2,    a^-\-y^  =  S2. 
33.  a;  +  //  =  1072,  rc^/^  +  ^/^  =  6.        34.  re  -  y  =  a,  x*  +  i/  =  bK 

35.  dx-2y  =  13,     (x  +  yY/^  +  2{x-yY^'=d{ar'-yYK 

36.  7x  +  Dy  =  29,     (2x  +  y)/{Sx-y)-(x-'y)/(x  +  y)  =  ^8/15. 

37.  5a;-7?/  =  4,    (a^  +  f)/(x  +  2jy+{3^-f)/(x-yf  =  4dx/^, 

38.  a;  +  ?/  =  2,      13(:r-h?/^)  =  121(:c3  +  ?/3)^ 

39.  x  +  y  =  4,     4l(a^-{-y')  =  122(x'  +  y'). 

40.  a:  +  y  =  «,     2:/(Z>  — ?/)  +  (^-2/)/a;=c. 


208  QUADRATIC   EQUATIONS.  t^Tl,  PR 

PrOB.  4.   To  SOLVE  A  PAIR  OF   QUADRATIC   EQUATIONS   IN- 
VOLVING  THE   SAME    TWO    UNKNOWN    ELEMENTS. 

No  one  rule  is  best  for  all  cases;  many  special  devices  may 
be  used,  and  the  examples  given  below  suggest  methods. 

If  by  combining  the  old  equations,  new  equations  can  be 
found  that  involve  some  of  the  familiar  type-forms,  such  as 
'Z^±2xy  +  y'^,  x^  —  y^y  then  very  often  either  the  square  root 
may  be  found,  or  by  factoring  and  division  a  quadratic  equa- 
tion may  be  replaced  by  a  simple  one. 
E.g.,  if  3a;.y-4a;-4^  =  0,  x^^y^^-x^-y-1^^^\ 
put     {x  +  ijY-2xy    for    x^  +  if,     and  write  the  equations 

Zxy-4:{x  +  y)  =  0,     (x  +  yY  +  {x+y)-'ixy-'^Q  =  0, 
eliminate  xy,  and  solve  the  quadratic  in     {x-{-y)\ 
tlien  aJ-h?/  =  6,    xy-Sy    or    x  +  y=~l^/Z,    xy=-62/9; 
solve  these  two  pairs  of  equations  for  x,  y. 
So,itx-y  =  ixy,        x^  +  y^  =  ixy: 
subtract  the  square  of  the  first  equation  from  the  second  and 

solve  the  resulting  equation  to  find  the  values  of  xy; 
join  each  of  these  equations  with  the  first  equation  to  find 
values  of  x,  y. 

So,  if  x^-hxy  +  if-=ld,      x^-xy  +  2f='7: 

find  the  value  of  xy,  then  of    x  +  y,    x-y,     then  of    x,  y. 

There  are  four  pairs  of  roots. 
So,  if  i/{x  +  y)  +  ^(x -y)=  y/a,       -,/(x« -f  f)  +  ^(x^^ -y')  =  b: 
then     2x  f  2 ^(x^ - y^)  =  a,     2o^-^'Z .^{x^ -?/*)  =  ^^      [squaring, 
and      x''-y^=i(t'~ax-vA     3^-f=ih^-h^3?-\-xK 
i.e.,      f-ax-iaK  y*=^V-iJ*; 

,\b^x^  —  ib*  =  {ax—ia^)\    whence  x  is  found;  then  y. 
So,  if  {x-\-ijy-2xy=r-{x-hy)+2G,     Qxy^-H(x^y): 
then    {x  +  yY-l{x  +  y)  =  2Q;  [elim.  xy. 

and      x  +  y  =  Q    or     -13/3,    xy  =  S    or     -52/1); 

X,  y  =  4,   2;   2,    4;     i(-13±i/377),     1{-U^  i/277). 


4>§3]  TWO   UNKNOWN  ELEMENTS.  209 

QUESTIONS. 

Solve  the  joair  of  equations: 

1,  ix"'  +  7^'  =  148,  3.6-^ - ^2  =  1 1.       2.  x  +  ^  =  x\  3y  - x  =  if. 
3.  x^^ifz^^xij,     x-y-\xy.          4.  y^^xy^^,   x^  +  y'^=b, 
5.  x^  +  y^  =  a^,     xy  =  l)\  6.  Q?  +  y^  =  9,     xy-2. 

7.  0?  +  xif  =  10,     ?/3  +  x^y  =  D.  8.  x^  +  y  =  4:X,  y^  +  x  =  4y. 

9,  a;%c +  ,?/)  =  SO,     x%2x-3y)  =  80. 

10.  :6-^  +  a;y  +  f/*=:133,     x^-xy  +  y'=7. 

11.  i^=ax  +  dy,     y^  =  ay  +  I?x. 

12.  :c  +  ?/=10,     -j/r^'^-^+V^.?; -1  =  5/3. 

13.  bx  +  ay  =  aby     hx-\-ay  =  'ixy. 

14.  82.^/^-^1/2=14,     .^^/y/2^2/. 

15.  81a;*- 16/=  1296,     9a.-2  +  4/  =  36. 

16.  a.-2  +  :^'?/  =  63,   ?/2  +  ^?/  =  18.        17.  a;-?/  =  9,   (^-f=2^'^. 
18.  .^'2-a;^/  +  ?/2  =  25,  a.'^  +  ?/3=125.     19.  2{x  +  y)  =  lo,  xy  =  Q. 
20.  4a;  +  4z/  =  12,     it'^  +  ?/3=63.  21.  x-^y^S,     xy  =  4. 
22.  a;3  +  «/^  =  3(a;  +  ?/),     x-y=zl,       23.  2;2  +  2^a=i3^  -a;?/  =  6. 

24.  x^  +  y'^  =  25,     xy-x  +  y--b, 

25.  4(:z;  +  «/)  =  8rc^,     a;  +  ?/  +  a;2^^a_26, 

26.  x-^y  +  i/xy=14:,     x^  +  y^  +  xy  =  S'l. 

27.  2;^  +  6.^  +  7y  =  50,     dxy  +  2x  +  5y  =  72. 

28.  ^'*  +  a:2/  +  «/*=243,     a?-xy  +  y^  =  21, 

29.  :c*-2;3y  +  2:2^2_^^3^^4_31^     ar*  +  t^  =  31, 

30.  a;2  +  yH2:  +  2^  =  4,     2a:?/  +  3:r  +  3^  =  8. 

31.  x-y-2^{X'-y)=-\,     7?-y^  +  4:\^{x^-f)^Q0, 

32.  2;  +  2/  =  5/6,     x^y^  +  l/a:^^^ ^ ^^y ^  ^/xy)  -  60||. 

33.  Sxfy  =  bOy/x,     xy-\-x-y  =  l^, 

34.  :?^/2/^  +  {2x  -t-  2/)/t/2^  =  20  -  (y^  +  rr)/^,     r?;  +  8  =  4^. 


210  QUADRATIC  EQUATIONS.  [Vll,  Pr. 

CHANGE   OF  THE   UNKNOWN   ELEMENTS. 

Sometimes  the  solution  of  a  pair  of  equations  may  be  sim- 
plified by  changing  the  unknown  elements. 
E.g.,  if    x  +  y  =  4:,     a^  +  y'=^82: 
put      u-{-v    for  X,     u-v    tor  y; 
then     (u  +  v)  +  (2i-v)  =  4,     ti  =  2, 
and      {u  +  vy+{u-vy  =  82,    «^H6?^V  +  z;*  =  41, 

v^  +  24:V^-25  =  0, 

1/^=1     or     —25,     v=zhl     or     ±54/"!, 

a;  =  3,     1,     2  +  54/-I,     2-5|/-l, 

y  =  l,     3,     2-5|/-l,     2  +  5|/-l. 
So,  if  a;*-a:2  +  /-/=84,     aH:tV  +  ^'=49: 
then     {ar^+yY-2xY-{x'  +  y')  =  ^^.     (x'' +  f)  +  xy  =  4<J ; 
put  ?^  for    a^+y%     v  for    ic^/,     and  solve. 
There  are  eight  pairs  of  roots. 

BOTH  EQUATIONS  OF  THE  FOKM  ax^ +  hxy  +  cy'^^Tc, 

If  both  equations  be  of  the  form  ax^  +  bxy  +  cy'^  =  h,  then: 
replace  y  by  vx,  eliminate  x^  by  co7npariso)i,  and  solve  the 

resultant  for  v; 
replace  v  by  Us  values  in  either  of  the  vx-equations  and  solve 

for  x; 
get  the  products  of  corresponding  values  of  v,  x  for  values  ofy. 
Check.  Replace  x,  y  by  their  pairs  of  values  in  the  other 
original  eqtiation,  and  see  whether  it  be  satisfied  thereby. 

E.g.,  if  22;?/  +  5/=195,     2:)?-Axy=il: 

then  •/  2vx^  +  bv'^x^=  195,     Zx^ -4vx^=l,  [repl.  y  by  vx, 

.-.  7(2y  +  5r'^)  =  195(3-4i')     and     v  =  b/l     or     -117/5; 
.\^x'~^^^x^  =  l,     a;=dz7,     ?/=±5,     if     r  =  5/7. 
and     32;*  +  ^|-aar^=7,    a;=  ±  |/(5/G9),     y=^U1!/\/Ub     if 
v=  —  \  17/5,     four  pairs  of  roots. 
Check.  2.7.5  +  5.52=195,     2- -7- -5  +  5-(-5)''=  195; 
and  so  for  the  other  pair  of  roots. 


4.  §2]  TWO  UNKNOWN   ELEMENTS.  211 

QUESTIONS. 
Solve  the  pair  of  equations: 

1.  x  +  y  =  b,     x'^  +  if^^l.      2.  x'^  +  ^xij  =  b^,xy  +  ^ifz^  lib. 
3.  ^-.?/  =  3,     a;-^-^^=:3093.         4..  x^/^^-if/'=l,  x^  +  y=ll. 

5.  ^x"  +  xy  +  4?/2  =  6,     'dx"  +  8?/^  =  1 4. 

6.  a;*  +  ^*=14a;y,     a;  +  ?/  =  9.       7.  x^-y^  =  a\     xy  =  h\ 

8.  a;H/  =  45,  2:  +  ?/  +  |/2.v;^  =  15. 

9.  x'  +  y''=l+xy,     x''  +  y''=Qxy-l, 

10.  re2_.^;^^2  +  ?/2  =  2,     a;*4-?/'  =  20. 

11.  xy{x  +  y)  =  '6<d,     x^  +  y^=^b. 

12.  a;2^  +  .T/  =  48,     xhj -xif=ilQ. 

13.  a;  +  ?/4-l  =  0,     :?;^  +  ^H211  =  0. 

14.  oy'  +  f  +  xy^lb^,     x^-y^=2l. 

15.  l/:ci/3-l/y/^r=l,     1^-1/?/ =  37. 

16.  xyy  +  yyx  =  ^/'2,     'i/{x  +  y)  =  l. 

17.  .^•2  +  xy  +  2^2  =  74,     2.^-2  +  2.r?/  +  ?/  =  73. 

18.  :6'*-a,-2  +  ?/*-^2^84,     x^  +  x'^if  +  y^^A'd. 

19.  :?;^  +  ?/3  +  3a;  +  3?/  =  378,     a-^  -\-y^-  'ix  -  3//  =:  324. 

20.  x^-\-if  +  x-{-y  =  bO,     xij-+x^-y  =  2^. 

21.  a;*-a;y  +  i/*  =  16,     2a;V3a;y-3/=32. 

22.  a;2  +  «'  =  2/'  +  ^'=G'C  +  ?/f+(r^-Z')l 

23.  Qy^y  +  xif~^0,     a:y+uy  =  468. 

24.  a;Ha;?/  +  ?/^  =  84,     re -y.r?/ +  ?/ =  6. 

25.  .r^-//  =  240,      {x  +  yy  =  ^Q. 

26.  a;"*2/^=(3/2)"'-»,     a;^?/"^=(2/3)'"-^ 

27.  |/CV2/)  +  |/(2/A)  =  10/3,    a;  +  ?/  =  10. 

28.  xyy^  +  ?//a;  +  a://y  =  27/4 - y^/x\     x-y  =  2. 

29.  (a;  +  ?/)07.^-?/)  =  819,     (:c^?/)(.t3  +  ?/3)  =  399. 

30.  a;*  +  a.Y  +  3/*  =  931,     a;2-a:?/  +  /=19. 


212  QUADRATIC   EQUATIONS.  [VII,  Pr 

§  3.    THREE   OR  MORE   UNKNOWN   ELEMENTS. 

PkOB.   5.    To  SOLVE  A  SYSTEM  OF  71  QUADRATIC  EQUATIONS, 
INVOLVING   THE    SAME    U   UNKNOWN    ELEMENTS. 

The  examples  given  below  suggest  methods. 
:Ei.g.,itx{x  +  y-\-z)  =  18,     y{x-\-y  +  z):=  12,     z{x  +  y-{-z)  =  6: 
then     (x-}-y  +  z){x  +  y-\-z)  =  36,  [add. 

x  +  y  +  z=  ±6,  [take  sqr.  rt. 

and      ^=±3,    y=±2,    z=  ±1.   [div.  eqs.  1,  2,  3  by  :i;  +  7/  +  2;. 
So,  if  xyz  =  (r{y  +  z)  =  ¥{z  +  x)  =  c^(x  +  y) :  divide  by  xyz-y 
then     1  =  a\l/zx  +  1/xy)  =  h\l/xy  +  1/yz)  =  c\l/yz  +  1/zx), 
and      1/yz  =  \{-l/(t'  + 1/¥  +  l/d"), 
l/zx  =  l{\/a'^-l/h'-^l/v% 
l/xy  =  h{\/a'  +  l/W-l/c'y, 
and  *.*  x^=l/yz:  (l/zx-l/xy),  [identity. 

.\x^  =  2{-l/a^^l/b''  +  l/c') 

:  [l/«2  -  1/b''  +  1/r) .  (1>^  +  1/Z/^ - 1/6-2)]; 
and  so  for  y"^,  z^. 

So,  if    y^  +  yz  +  z^  =  l,     z''  +  zx  +  x^  =  lZ,     'x'^  +  xy +  y^=^: 
then     (.T  — ?/)(a;  +  2/  +  2;)  =  6,  [sub.  eq.  1  from  eq.  2. 

{z  —y){x  +  y-hz)  =  10.  [sub.  eq.  3  from  eq<  2. 

.*.  x-y:z~y  =  '^:5,    6x-5y-3z-'dy,    x=l{2y-\-3z); 
.•.e^  +  K'^y  +  3.).i(27/  +  8.)  =  13; 

[repl.  X  by  l{2i/-\-?vy)  in  eq.  2. 
combine  this  equation  with  eq.  1,  and  solve  for  //,  z; 
then     ij=±2,     ±1/|/19;     z=zqz'S,     ±11/|/19; 
and      .2:=  T=l,     ±7/4/19.  [repl.  //  in  eq.  3. 

So,  if    x^  —  yz  —  a,     y^  —  zx  =  h,     z^  —  xy  =  c: 
from  the  square  of  each  equation  subtract  the  j)roduct  of  the 

other  two; 
then  •.  x{x^  +  y^  f  z^  —  3xyz)  —  d^  —  be, 
y{a?  -\-y^-\-^-  Sxyz)  =  b^-  ca, 


5,  §3 J  THREE   OR  MORE   UNKXOWX   ELEMENTS.  213 

and      z{3?  +  2f  +  ^  —  2>xyz)  =  i^  —  ah. 

.-.  :c/{a'  -  be)  =  y/{b'  -  ca)  =  z/{c'  -ah), 

.-.  x=  ±  (cr  -  ic) / \/(a^  +  b^ -\-  c^  -  dahc).     [repl.  i/,  z  in  eq.  1. 
So,       y=±  {b^  -  ca)/i/{a'  -^P  +  (r-  3abc), 
z=±  {r - ab)/ ^{a^  +  P  +  c^ -  '6abr\ 

QUESTIOXS. 

Find  the  values  of  :r,  y,  z  from  the  set  of  equations: 

1.  yz  -be,        2.  xyz   =]0,  3.  :c^y-^z  =3^, 

Ix^-ay-aly  yz/u-=  10,  x'^-hy-'^ +  z-'^  =  3^, 

ex  +  az  =  ac.  -'^y/^  =  10.  o-yz  =  1 . 

4.  ^x  +  ij-Sz    =0,     5.  xy  =  a{:r+ii),       Q,  :r-^y^z      =6, 
4;/:  -  8^' +  7^  =  0,         xz  =b(x  +  z),  4:X  +  y-2z  =  0, 

yz-\-zx  +  xy  =  47.        yz  =c(y-\-z).  o^  +  y'^  +  z^  =14c 

t.x  +  y  +  z  =13,  S.x  +  2y-z  =11,  9.  3a-  +  ?/-2^  =  0, 
2-2  +  y^  +  ^2  ^  65,  :r  -  4if  +  z''  =37,  4./;  -  y  -Bz  =  0, 
xy  =ia        xz  =24.        x^  +  f+^=467. 

10.  x  +  y  +  z     =a,  11.  x-\-y  -z, 

x''  +  y^  +  z^  =  a\  3.r-2//  +  17^  =  0, 

s.-^  +  y^  +  ^  =  «\  a^  +  3^^  +  2z^  =  167. 

12.  (x-{-y)(y-\-z)=30,         13.   (?/-5;)(2;  +  :r)  =22, 
(2:  +  ^)(?/  +  2-)z=15,  (2;  +  2:)(a:--?/)  =  33, 

{z  +  x)(z  +  2j)=  IS.  (x-y)(y-z)  =  6. 

14.  a:^^ V?/'  =  12,  15.  (x  + 1')  (j/  +  2)  =  -  «  i-  ^  +  c, 

x^y^z  2V^  =  8,  ( V  +  ?/)  ( .r  +  ^)  =  <-/  —  Z/  +  <7, 

x^yzV  =  1,  («^  +  z)  {x  +  iy)  =  rM-  ^  +  f , 

xyhV  =  4/3.  a;^  +  ?/H  z'  4-  i^  =  3  (r/  f  ^  +  r). 

16.  xyz/(x  +  y)=-8,  17.  a-?/V  =  108,     18.  a-^/^  =12, 

xyz/iy-\-z)=24,  yz^x  =  18,  xh/z^  =54, 

xyz/{x  +  z)  =  12.  y^ry^  =  2/3.  x''y^z^=  72. 

19.  3a:2^2i/H52;2  =  0,      7.^-^     3/- 15^^  =  0,     bx-Ay-\-lz^<6. 


214  QUADRATIC  EQUATIONS.  [Vn. 

§  4.    QUESTIONS   FOR  REVIEW. 
Define  and  illustrate: 

1.  A  complete  quadratic  equation. 

By  what  other  name  is  such  an  equation  known? 

2.  An  incomplete,  or  pure,  quadratic  equation. 

Give  the  general  rule,  with  reasons  and  illustrations,  for: 

3.  Solving  an  incomplete  quadratic  equation. 

4.  Solving   a   complete    quadratic   equation    of   the   form 

x^-i-px  +  q  =  0;     of  the  form     ax^-{-ljx-\-c  =  0. 

5.  Solving  a  pair  of  quadratic  equations  involving  the  same 
t\A'o  unknown  elements,  when  one  equation  is  simple  and  the 
other  quadratic;  when  both  equations  are  quadratic. 

G.  Discuss  the  equation  x^+px  +  q  =  0  :  what  are  the  roots 
it p  be  positive  and  q  be  negative  ?  if  p,  q  be  both  negative? 
if  JO,  q  be  both  positive  ?  if  ^^  be  negative  and  q  be  positive  ? 

Find  the  sum  of  the  roots,  and  their  product. 

Show  what  relation  between  p  and  q  makes  the  two  roots 
equals;  opposites;  reciprocals;  opposite  reciprocals. 

7.  Discuss  the  equation  ax^-{-bx-i-c  =  0  :  what  are  the 
roots  if  c  be  zero?  if  b  be  zero  ?  if  «  be  zero? 

What  relation  have  a,  b,  c,  if  the  two  roots  be  real  and 
equal  ?  if  real  and  unequal?  if  imaginary? 

Can  one  root  be  real  and  the  other  imaginary  ? 

8.  Show  how  to  form  a  quadratic  equation  that  shall  liave 
two  given  numbers  for  its  roots. 

9.  If  cr,  /^  be  the  roots  of  the  equation  ax^  +  bx  +  c  =  0, 
find  the  value,  in  terms  of  a,  b,  c,  of  a- (3,  a^-\-p^y  a^  —  p'\ 
a'  +  p\     a'-p\     a/fi^fi/a,     a'/ (i-^  (iy  a. 

10.  So,  find  the  equation  wlioso  roots  are : 

a/fi,     /?/.r;     a/fi,      -/?/..;     a\     fi'^-     \/a\     \/(i\ 

11.  Show  how  to  solve  a  quadratic  equation  by  factoring. 

12.  Show  how  to  factor  a  quadratic  function  by  solving  the 
quadratic  equation  formed  by  [)utting  the  function  equal  to  0, 


$4]  QUESTIONS   FOR   REVIEW.  215 

13.  Show  how  a  pair  of  equations  that  involve  the  same  two 
unknown  elements  may  sometimes  be  solved  by  changing  the 
unknown  elements. 

14.  Solve  the  equation  2x'-9x^-i-U3:'^-9x-\-2  =  0. 
[Divide  by  .T^;  then  2{x''-hx-')-9(x  +  x-')  +  U  =  0; 
write  y  for  x-^x-'';     then     2(?/2-2)-9^  +  14  =  0  ; 

solve  tliis  equation  for  y,  then  the  equation  x  +  x-^  =  y  for  x. 

Such  equations,  in  which  terms  equidistant  from  the  ends  of 
the  function  have  equal  coefficients,  are  7'ecijj?'0cal  equations, 
and  their  roots  come  in  reciprocal  imirs. 

THE    ROOTS    OF   +1    AXD    OF   "1. 

15.  The  three  cube  roots  of  +1  are  1,  ^(  — ld=|/  — 3). 
[Write     x=l/l;     then    x^-l  =  0={x-l){x^  +  x  +  l)', 

put  these  two  factors,  in  turn,  equal  to  0,  and  solve  the  equa- 
tions so  formed  for  x. 

16.  The  sum  of  the  three  cube  roots  of  +1  is  0,  and  so  is  the 
sum  of  their  products  in  pairs;  the  product  of  the  last  two 
roots  is  "^  1,  and  each  of  them  is  the  square  of  the  other. 
Express  the  three  cube  roots  of  1  by  1,  r,  r^,  and  show  that: 

17.  l+r  +  r^^O.         18.   (l-r  +  r^)(l  +  r-r2)=r4. 

19.  {l-^r''Y  =  r\         20.   (l-r)- (l-r^)  .(l-r*).(l-r^)  =  9. 

21.  {l-r  +  i^)-{l-r''  +  r')-{l-r'  +  r^)"-2}i  factors  =  2''". 

22.  Find  the  three  cube  roots  of  "1. 

The  sum  of  these  three  roots  is  0,  and  so  is  the  sum  of  their 
products  in  pairs;  and  either  of  the  last  two  roots  is  the  oppo 
site  of  the  square  of  the  other. 

23.  What  are  tlie  cube  roots  of  '^a^  and  of  ~a^  ? 

24.  Find  the  four  fourth  roots  of  +1. 

The  sum  of  these  four  roots  is  0,  and  so  are  the  sums  of 
their  products  in  pairs  and  in  threes;  and  the  product  of  the 
four  roots  is  "1. 

25.  The  four  fourth  roots  of  - 1  are  (1  +  /)  •  ^^,  (1  -  i)  •  ^i, 
(l  +  i)--i/i,     (l-i)--Vi.  "       [/=|/-i. 

26.  What  are  the  fourth  roots  of  ^a*  and  of  ~a*? 


216  QUESTIONS  FOR  REVIEW.  {VII. 

27.  The  five  fifth  roots  of  +1  are 

1,    i[-l±|/5±?V(10  +  24/5)]. 
[Write  .T^-l  =  0  =  (.r-l)-(.T*  +  r6-^  +  :f2  +  rr+l); 
then     x-\  =  0     and     x  =  l: 

and    .•  x'-]-x  +  l  +  x-'^  +  x-^  =  0;     [put  sec.  fact.  =  0,  div.  by  x\ 
.-.  x''-\-2-\-x-''  +  x  +  x-''=\,      {x-hx-'Y+{x-\-x-')+i  =  i, 
.'.  x-\-x~'^=  — 1  +  4^4/5;     and   these   two  quadratic  equa- 
tions, when  solved,  give  the  last  four  values  above. 

The  sum  of  these  five  roots  is  0,  and  so  are  the  sums  of 
their  products  in  pairs,  in  threes,  and  in  fours;  and  the  prod- 
uct of  the  five  roots  is  ~1. 

28.  Find  the  five  filth  roots  of  "1. 

29.  What  are  the  fifth  roots  of  '^a^  and  of  ~a^? 

30.  The  six  sixth  roots  of  +1  are  the  three  cube  roots  of  +1 
aud  the  three  cube  roots  of  ~1. 

Group  these  six  roots  in  three  pairs  of  opposites. 

31.  The  six  sixth  roots  of  "1  are  ±i,     +|(|/3±/). 
[Write    x^-\-l  =  0=  (x^  +  l)(x^-x^  +  l);    put  these  two  factors, 

in  turn,  equal  to  0,  and  solve  the  equations  so  formed.] 

32.  What  are  the  sixth  roots  of  -"a^  and  of  -(i^  ? 

MAXIMA   AND    MINIMA. 

33.  If  X  vary,  but  remain  real,  what  is  the  least  possible 
value  of    x^  —  4:X-\-3  ? 

[Write     a;2-4.r  +  3  =  ?/;     then     {x-2Y  =  y  +  l, 
and  •.'  (x  —  2Y     is  always  positive, 

,'.  yfl     IS  positive  and  ~1  is  the  value  sought. 

34.  Find  the  least  value  possible  of  x^-\-px-\-q,  and  find 
the  value  of  x  which  makes  this  function  least. 

35.  If  :/;  vary,  but  remain  real,  what  bounds  has  the  frac- 
tion (.rH 2.6- -11)7(2: -3)  ?  what  are  the  like  values  of  x? 
[Write  (x'  +  2x  - 1 1  )/(.r -  3)  =  // :  then  x'  +  (2  -  i/)x  +  3 //  - 1 1  =  0, 
and  %•  X  can  be  real  only  when     (2  — ?/)'^<12^  — 44, 


§41  QUADRATIC  EQUATIONS.  217 

Le.f      when   i/^-16// f  48<0   and  (?/-4)(//- 12)   is  positive, 
.*.  the  fiaction  has  no  valne  between  4  and  12,  but  has  all 
otlier  values. 

Equate  the  fraction  to  4  and  12,  in  turn,  and  solve  for  x. 

36.  The  fraction  [x  +  a) / [x^  +  bx  +  c^)  lies  always  between 
two  fixed  bounds  if  (i^-i-c^>ab  and  Z'^< 46-^;  it  lies  always 
beyond  two  fixed  bounds  if  a^-hc^>ab  and  b^>4:c^;  and 
it  takes  all  values  if     a^-\-v^'<ab. 

In  the  light  of  ex.  3G,  find  the  bounds  of  these  fractions: 

37.  (x  +  'k)/(x^  +  2x-{-8).  38.  (x  +  2)/(x'-h8x-\-4). 
39.  (a;  +  8)/(.r2  +  4.T  +  2).  40.  {x  +  4)/(x''  +  8x  +  2), 
41.  {x  +  2)/(x^-{-4:X-\-S).              42.   (x  +  8)/(x'  +  2x  +  4), 

43.  Divide  a  given  number  a  into  two  parts,  (a)  whose 
pi-oduct  shall  be  a  maximum,  (b)  the  sum  of  whose  squares 
shall  be  a  minimum. 

44.  Of  all  the  rectangular  fields  that  have  the  same  area,  the 
square  has  the  shortest  perimeter,  and  the  shortest  diagonal. 
[Use  the  equations 

(x^yy^{x-yy^4rxi/,     x' +  if  ={x-yY +  2xy. 

45.  Of  all  the  rectangles  that  have  the  same  perimeter,  the 
square  has  the  gTeatest  area.     Which  of  them  has  the  least  ? 

46.  Find  the  condition  that  the  quadratic  function 
ax^  +  2hxy  +  by-  +  2gx  +  2fy  +  c 

may  be  resoh^ed  into  two  linear  factors. 
[Put  the  function  equal  to  0  and  solve  the  equation  for  x  ; 
then     ax  +  hy-\-gz=z  ±  ^J  [jf{h'  -  ab)  +  2y{h(i  -  af)  +  (/  -  ac)\ ; 
and  this  radical  is  rational  only  if  {lig  —  afY  =  (h^  —  (tb)(g''  —  ac), 
i.e.,  if    abc-\-2fgli  —  ap  —  bg'  —  c¥  —  ^. 

In  the  light  of  ex.  46,  show  whether  these  functions  can  be 
factored,  and,  if  possible,  factor  them. 

47.  5a;' 4- \2xy - 9//' -2ix~\8y-h. 

48.  hx^-V\2xy-^''-2^x^\8y^h. 

49.  hx'-\8xy^^y''^2^x-\2y-h. 

50.  5a:' -18:^;^  4-9/4- 24a; +  18?/ +  5. 


'2lS  PROGRESSIONS,  INCOM.  POWERS,  LOGARITHMS.  tVin,TH8. 

VIII.  THE  THREE  PROGRESSIONS,  INCOMMEN- 
SURABLE POWERS,  AND  LOGARITHMS. 


A  series  is  an  expression  such  that  each  term  is  connected 
with  the  preceding  terms  by  some  law.  [comp.  df.  series,  p.  34. 

§1.  ARITHMETIC   PROGRESSION. 

A  series  is  in  aritlimetic  progression  if  each  term  after  the 
first  be  found  by  adding  a  constant  to  the  term  before  it.  The 
constant  added  is  the  common  difference.  A  series  in  arith- 
metic progression  is  an  arithmetic  sei'ies. 

The  abbreviations  are:  a  first  term,  I  last  term,  d  common 
difference,  n  number  of  terms,  s  sum  of  all  the  terms. 

If  d  be  positive,  the  series  is  a  rising  series;  if  d  be  nega- 
tive, it  is  2k  falling  series. 
E.g.,  1,  3,  5,  7,  9     is  a  rising  series, 

wherein     ^=+2,     a  —  1,     /=9,      ?i  =  5,     5  =  25. 
So,     9,  7,  5,  3,  1,  "1,  "3,     is  a  falling  series, 

wherein     6?=  ~2,     a  =  9,     ^=~3,    ?i  =  7,    5  =  21. 

Theor.  1.  In  a7i  arithmetic  scries,     l  =  a  +  {}i  —  l)d. 
For -.*  a,     a  +  d,     a-\-2dy     a  +  Zd,-",     a  +  {lc-\)d    are  the 
first,  second,  third,  fourth,  •  •  -^'th  terms,  [df. 

.'.  a  +  {n  —  \)d—  I,     the  last  of  a  series  of  n  terms.   Q. e.  d. 
Theor.  2.  In  an  arithmetic  series,    s  =  in{a-hl). 
For-/    s  =  a-^{a  +  d)  +  (a  +  2d)-{-'  "  -^{l  -d)-\-I,     ?i  terms, 
and        s  =  l-h(l—d)  +  {l -2d)+ ■  " +(a  +  d)  +  a,    ?/ terms, 
.-.  2s=(a  +  l)-\-(a-\-l)  +  {a  +  l)+  '"-h  (a  + 1),       n  times, 

.'.    s  =  \n{a-\-l).  Q.E.D. 

Note.  The  numbers  a,  d,  I,  n,  s  are  the  five  elements  of  an 
arithmetic  series.  In  theors.  1,  2  these  elements  are  connected 
by  two  equations;  hence,  any  three  of  them  being  given, 
the  other  two  can  be  found,  and  in  all  tliere  are  twenty  equa- 
tions :  four  that  contain  no  a,  four  that  contain  no  d,  and  so  on. 


1,2,§1]  ARITHMETIC  PROGRESSION.  219 

QUESTIONS. 

1.  Write  an  arithmetic  series  in  which  rt  =  3,  d=  ~2,  n  =  5. 

2.  What  two  conditions  would  make  negative  all  the  terms 
of  an  arithmetic  series?  Can  an  arithmetic  series  be  an  end- 
less rising  series  and  have  some  terms  negative?  all  negative? 

3.  From  the  equation  l  =  a-\-(?i  —  l)d,  find  the  value  of  a 
in  terms  of  I,  n,  d. 

4.  So,  solve  this  equation  for  w,  and  for  d. 

5.  Solve  the  equation     s  —  \}i{(t-\-l)^     in  turn  for  n,  a,  l. 

6.  In  the  series  of  integers  1,  2,  3,  •  •  •,  find  the  last  term 
and  the  sum  of  5  terms;  of  20  terms;  of  Ji  terms;  of  2/^  terms; 
of  271  +  1  terms. 

7.  In  the  series  of  odd  positive  integers,  find  the  sum  of  35 
terms;  of  50  terms;  of  n  terms;  of  2?i  terms;  of  2/^  +  1  terms. 

8.  In  the  series  of  even  positive  integers,  find  the  sum  of  /t 
terms;  of  2ii  terms;  of  2;i  — 1  terms;  of  2w  +  l  terms. 

9.  Find  the  five  elements  of  the  arithmetic  series 

1,  3,  5,..-  99;     1,  3,  5,•..2^•-l;     4  +  5+ ...  =5350. 

10.  In  an  arithmetic  series,  tlie  sum  of  two  terms  equidis- 
tant from  the  extremes  is  constant. 

11.  AVhat  multiple  of  the  common  difference  must  be  added 
to  the  fifth  term  of  an  arithmetic  series  to  give  the  ninth  ?  to 
the  twelfth  to  give  the  twentieth  ? 

12.  If    a  =  l,     d=2,     then     s  =  n\ 

13.  A  three  digit  number  is  26  times  the  sum  of  its  digits, 
which  are  in  arithmetic  progression;  if  396  be  added  to  the 
number  the  order  of  the  digits  is  reversed,  find  the  number. 

14.  A  hundred  stones  are  laid  in  a  row,  a  meter  apart,  and 
a  basket  is  placed  a  meter  from  the  first  stone:  how  many 
kilometers  will  a  man  walk  who,  starting  from  the  basket, 
picks  up  all  the  stones,  one  by  one,  aiid  brings  them  separately 
to  the  basket  ? 

15.  The  sum  of  three  numbers  in  arithmetic  progression  is 
27  and  the  sum  of  their  squares  is  293 :  find  the  numbers. 


220     PROGRESSIONS,  INCOM.  POWERS,  LOGARITHMS.  LVII1,Pr. 


ARITHMETIC   MEANS — INTERPOLATIOiT. 
PrOB.  1.    To  INTERPOLATE  Nl  ARITHMETIC  MEANS  BETWEEN 
TWO   NUMBERS  a,  I 

Divide  the  remainder,    I— a   hy  m -\-l  for  the  common  differ- 

ence;  to  a  add  one,  tioo,  three'  •  -times  this  difference. 
E.g.,  to  interpolate  5  means  between  12  and  48: 
then  •.*  (48  — 12) :  (5  + 1)  =  6,     the  common  difference, 
.-.  the  series  sought  is     12,  18,  24,  30,  36,  42,  48. 
Note.  By  aid  of  this  problem,  from  every  arithmetic  series 
a  new  arithmetic  series  may  be  formed  by  interpolating  the 
same  number  of  arithmetic  means  between  every  pair  of  con- 
secutive terms;  and  the  common  difference  of  this  new  series 
is  the  quotient  of  the  common  difference  of  the  other  divided 
by  one  more  than  the  number  of  terms  so  interpolated. 
E.g.,  if  two  means  be  interpolated  between  pairs  of  consecu- 
tive terms, 
then  the  series  6,  12,  18,  24, 

becomes  the  series     6,  8,  10,  12,  14,  16,  18,  20,  22,  24. 

geometric  illustration. 
Let  ox,  oy  be  two  straight  lines  at  right  angles  to  each  other; 
take  points  a,  b,  c,  •  •  •  such  that  they  are  1,  2,  3,  •  •  •  inches 
above  ox,  and  6, 12, 18,  •  •  •  inches  to  the  right  of  oy; 

Y 


0  G  l;i  13     X 

then  A,  b,  c,  •  •  •  lie  on  a  straight  line  through  o,  and  their 
distances  from  ox  are  in  arithmetic  progression,  and 
so  are  their  distances  from  oy,  and  from  o; 

and  the  common  differences  of  these  three  series  are  1, 6,  |/37. 


1.  §n  ARITHMETIC  PROGRESSION.  221 

Between  a,  B,  c,  •  •  •  interpolate  other  equidistant  points; 
then  thres  new  arithmetic  series  are  formed,  however  close 
together  the  points  of  division  may  lie. 

COXTINUOUS   PROGRESSION. 

Lay  a  pencil  on  the  figure  and  move  it  slowly  to  the  right, 
keeping  it  always  parallel  to  OY  and  letting  it  cut  ab 
at  a  moving  point  p; 

then  three  series  are  formed,  in  co7itimious  arithmetic  prorjres- 
sion  :  the  growing  distances  of  p  from  ox,  from  oy,  and 
from  o. 

questions. 
Find  the  five  elements  of  the  arithmetic  series: 
1.  5,    6  means,  75.  2.  3,  6  means,  —11. 

3.  2|,  4  means,  20.  4.  5  terms,  19,  7  means,  67. 

5.  When  m  arithmetic  means  are  interpolated  between  two 
given  terms,  these  two  terms  and  the  means  make  a  series  of 
how  many  terms  ?     What  are  a  and  I  in  this  new  series? 

6.  Form  a  new  arithmetic  series  by  interpolating  three 
means  between  every  pair  of  terms  of  the  series     4,  8,  12,  •  •  • 

What  is  the  new  common  difference  ? 

Form  a  new  series  by  taking  every  fifth  term  of  the  series 
last  formed,  beginning  with  the  second. 
What  is  now  the  common  difference  ? 

7.  Show  that,  at  simple  interest,  the  given  principal  and  its 
amounts  for  successive  years  form  an  arithmetic  series,  wherein 
n  is  one  more  than  the  number  of  years.     What  are  a,  d,  I  ? 

Write  such  a  series  for  five  years,  and  show  that  the  final 
amount  agrees  with  the  formula     l  —  a  +  {n  —  l)(J. 

8.  If  the  interest  be  reckoned  half-yearly,  how  many  means 
must  be  interpolat(^d  between  every  two  terms  ?  if  quarterly? 
if  monthly?     What  is  (/ in  each  of  these  new  series? 

If  the  interest  be  reckoned  instantly,  what  kind  of  a  series 
results  ?  what  is  the  number  of  terms?  the  common  difference? 
What  two  elements  of  the  series  are  unchanoed? 


222   PROGRESSIONS,  INCOM.  POWERS,  LOGARITHMS.  [VIII.  Ths. 

§2.   GEOMETRIC   PROGRESSION. 

A  series  is  in  geometric  progj^ession  if  each  term  after  the 
first  be  found  by  multiplying  the  term  before  it  by  a  constant 
multiplier.  This  multiplier  is  the  common  ratio.  A  series  in 
geometric  progression  is  a  geometric  series. 

The  abbreviations  are:  a  first  term,  I  last  term,  r  common 
ratio,  n  number  of  terms,  s  sum  of  all  the  terms. 

If  r  be  larger  than  1,  the  series  is  ?i  rising  series;  if  smaller, 
?i  falling  series. 
E.g.,  1,  2,  4,  8,  16     is  a  rising  series, 

wherein     r=+2,     «  =  1,     Z=16,     ?i  =  5,     e<f  =  31. 
So,    1,  "2,  4,  "8,  16     is  a  rising  series, 

wherein     r="2,     «  =  1,     Z=16,     w  =  5,     s  =  ll. 
But  16,  8,  4,  2,  1,  1/2,  1/4     is  a  falling  series, 

wherein    r=l/2,     a  =  lQ,     1=1/4:,     7i  =  7,     5  =  31|. 

Theor.  3.  In  a  geometric  series,     l  =  ar^~^. 
For  *.•  a,  ar,  ar^,  •  •  •  «r*"*     are  the  first,  second,  third,  •  •  - 
kih  terms,  [df. 

,\  ar''~'^  =  l,     the  last  of  a  series  of  ><  terms.      q.e.d. 
Theor.  4.  In  a  geometric  series,     s=(a  —  rl)/{l  —  r). 
For  •.•  s   =a  -i-ar  +r«;'H  •  •  •  +//'-^  +  /r-^  +  /,  [df. 

.♦.  rs  =  ar-\-a7'^  +  ar^+  •  •  •  +//•"* +  ^       -\-lr, 
,\  s  —  rs  =  a  —  lr,     and     s  =  (a  —  rl)/{l  —  r).         q.e.d. 

Cor.  1.  In  an  infinite  falling  geometric  series,  I  is  indefi- 
nitely small;  and  the  value  of  s  is  indefinitely  near  to  the 
quotient     r//(l— r). 

Cor.  2.  A  repeating  decimal  equals  a  common  fraction  irhose 
numerator  consists  of  the  repeating  figures,  a)id  the  denomi- 
nator of  as  many  9^9  as  there  are  repeating  figures. 

E.g.,  the  decimal     .45,     i.e.,    .454545  •  •  •,    is  the  sum  of  the 

geometric  series     45/100  +  45/100«  +  45/1 00'  +  •  •  • 
and      6'  =  45/100/(l-v}o)  =  45/99. 


3.4,6  2]  GEOMETRIC   PROGRESSION.  223 

QUESTIONS. 

1.  If  in  a  geometric  series  the  first  term  be  positive  and  the 
ratio  a  positive  proper  fraction,  what  signs  have  the  terms  ? 
how  do  they  change  ?  if  the  ratio  be  a  negative  proper  fraction  f 

2.  Solve  the  equation     l  =  ar'^~^    for  «  and  for  r. 

3.  In  a  geometric  series  «=  ~3,  ?==  ~48,  7'=  ~2:  what  is  ?i? 

4.  Solve  the  equation    .<?=  («  — r/)/(l  — r)    in  turn  for  a,  r,  I, 

5.  Find  the  last  term  and  the  sum  of  10  terms  of  the  series 
of  integer  powers  of  ^2;  oi  7i  terms;  of  2)1  terms. 

Find  tlie  12th  term  and  the  sum  of  12  terms  of  the  series: 
6.1-       +       _3/44----      7.  2/5  +  3/52  +  2/5^  +  3/5H••• 
8.  If  r  be  a  proper  fraction,  how  do  rising  powers  of  rvary? 
What  is  the  value  of  a  very  high  power  of  such  a  ratio  ? 

Find  the  value  of: 

9.  .212121...      10.  .672.      11.  .3684.      12.  .152727- .  • 

13.  The  geometric  mean  between  two  positive  numbers  lies 
between  them,  and  is  their  mean  proportional. 

14.  By  what  power  of  tlie  common  ratio  must  the  fifth 
term  of  a  geometric  series  be  multiplied  to  give  the  ninth 
term  ?  the  twelfth  term  to  give  the  twentieth  term  ? 

15.  In  a  geometric  series  the  product  of  any  two  terms 
equidistant  from  a  given  term  is  the  square  of  that  term. 

State  and  prove  the  like  truth  about  an  arithmetic  series. 

16.  If  all  tlie  terms  of  a  geometric  series  be  multiplied  (or 
divided)  by  the  same  number,  the  products  (or  quotients) 
form  a  geometric  progression  w4th  the  same  ratio  as  before. 

17.  From  the  two  fundamental  equations  l  —  af"^,  s  = 
{a  —  rl)/{l  —  r)f     eliminate  a,  I,  r,  in  turn. 

18.  A  man  invests  $100  in  stocks  that  pay  3  per  cent  half- 
yearly  dividends,  and  invests  the  dividends,  as  received,  at  the 
same  rate:  how  much  has  he  invested  at  the  end  of  5  years  ? 

19.  The  sum  of  three  numbers  in  geometric  progression  is 
13,  and  the  product  of  the  mean  by  the  sum  of  the  Qxtrem^s 
is  30;  what  are  the  numbers? 


224    PKOUKESSIONS,  INCOM.  POWERS,  LOGARITHMS,  ivm,  Pr. 


GEOMETRIC    MEANS. 
PrOB.  2.    To  INTERPOLATE  m   GEOMETRIC  MEANS  BETWEEN 
TWO   NUMBERS,  «,  I. 

Take  the  {m  +  l)th  root  of  the  quotient  l/a  for  the  commo7i  ratio; 
multiply  a  by  the  first,  second,  •  •  -  poioers  of  this  ratio. 
E.g.,  to  interpolate  three  means  between  3  and  48: 
then  •.•  ^^(48 :  3)  =  2,     the  common  ratio, 

.*.  the  series  sought  is     3,  6,  12,  24,  48. 

Note.  By  aid  of  this  problem,  from  every  geometric  series 
a  new  geometric  series  may  be  formed  by  interpolating  the 
same  number  of  geometric  means  between  every  pair  of  con- 
secutive terms;  and  the  common  ratio  of  this  new  series  is 
that  root  of  the  common  ratio  of  the  other  wliose  index  is  one 
more  than  the  number  of  means  so  interpolated. 
E.g.,  if  two  means  be  interpolated  between  pairs  of  consecu- 
tive terms, 
then  the  series        3,  6,  12  •  •  • 

becomes  the  series  3,     3-^2,     3-^4,     6,     Q>\/2,     C>\/^,     12-.- 

GEOMETRIC   ILLUSTRATION. 

Let  ox,  OY  be  two  straight  lines  at  right  angles  to  each  other; 
take  points  a,  b,  c,  •  •  •  such  that  they  are  0,  1,  2,  •  •  •  inches 
to  the  right  of  oy  and  3,  6,  12,  •  •  •  inches  above  ox; 

Y 


then  A,  B,  c,  •  •  •  lie  on  a  curve;  their  distances  from  oy  are 
in  arithmetic  progression,  witii  a  common  difference  1, 

but  their  distances  from  ox  are  in  geometric  progression,  with 
a  common  ratio  %, 


2,  §2]  GEOMETRIC   PROGRESSION.  225 

Between  a,  b,  c,  •  •  •  interpolate  other  points  whose  distances 
from  OY  are  arithmetic  means  between  the  terms  of 
the  series     1,  2,  3,  •  •  • 

and  whose  distances  from  ox  are  the  like  geometric  means 
between  the  terms  of  the  series     3,  6,  12,  •  •  • 

COXTIXUOUS   PROGRESSION". 

Lay  a  pencil  on  the  figure  and  move  it  slowly  to  the  right, 

keeping  it  always  parallel  to  OT,  and  letting  it  cut  the 

curve  at  a  moving  point  p; 
then  the  growing  distance  of  p  from  oy  forms  a  series  in 

continuous  arithmetic  progression, 
and  the  growing  distance  of  p  from    ox  forms  a   series   in 

continuous  geometric  p?'ogression. 

QUESTIONS. 

1.  From  the  formula  r=^~^{I/a)  find  the  new  ratio  when 
w?  geometric  means  are  interpolated  between  every  two  terms. 
Insert  geometric  means: 

2.  Four  between  1  and  32.        3.  Two  between  1  and  1000. 
4.  Three  between  1/9  and  9.    5.  Three  between  2  and  1/8. 

6.  Form  a  new  geometric  series  by  interpolating  three  terms 
between  each  pair  of  terms  of  the  series     3,  9,  27,  •  •  • 

What  is  the  new  common  ratio  ? 

Form  a  new  series  by  taking  every  fifth  term  of  this  series, 
beginning  with  the  second.     What  is  now  the  common  ratio  ? 

7.  In  compound  interest  the  principal  and  its  amounts  at 
the  ends  of  successive  years  form  a  geometric  series. 

Show  that  a=p{l-{-veitey  agrees  with  the  formula  I=a)-''-'^. 

8.  If  the  interest  be  compounded  half-yearly,  but  in  such  a 
way  as  not  to  change  the  final  amount,  how  many  means  are 
inserted  between  every  two  terms  ?  if  quarterly  ?  if  monthly  ? 

What  is  r  in  each  of  these  new  series  ? 
How  must  the  interest  be  compounded  to  make  the  amount 
a  continuous  variable  ?     What  is  then  the  value  of  7t  ?  of  r? 
AVhat  two  elements  of  the  series  are  unchanged  ? 


226  PROGRESSIONS,  INCOM.  POWERS,  LOGARITHMS.  [VIII.Th.5, 

§  3.   HARMONIC   PROGRESSION. 

A  series  is  in  liarmonic  j^rogression  if  the  reciprocals  of  the 
terms  be  in  arithmetic  progression. 
E.g.,    hhhh'"^     3,4,6,12,... 

The  last  term  of  a  harmonic  series  is  found  by  computing 
the  last  term  of  the  arithmetic  series  of  reciprocals  and  invert- 
ing it;  the  sum  of  a  harmonic  series  can  be  found  only  by  the 
actual  addition  of  the  terms. 

Theor.  5.    Of  three  consecutive  terms  of  a  harmonic  series 
the  first  is  to  the  third  as  the  excess  of  tlie  Jird   over  the 
second  is  to  the  excess  of  the  second  over  the  third. 
Let  p,  q,  r  be  three  consecutive  terms  of  a  harmonic  series; 
then    jj  :r=p  —  q:  fj  —  r. 
For  •.•  l/q  -l/p  =  \/r -  X/q,  [df. 

.-.  {p-q)/pq  =  {q-r)/qr 
and       {p-q)/{q  —  r)=pq/rq=p/r,  Q.E.D. 

PrOB.  3.  To  INTERPOLATE  m  HARMONIC  MEANS  BETWEEN 
TWO   NUMBERS,  a,  I. 

Find  m  arithmetic  means  hetween  the  reciprocals  of  a,  Z,  and 

take  the  reciprocals  of  these  means. 
E.g.,  to  interpolate  two  harmonic  means  between  12  and  48; 
then  •••  1/12  - 1/48  =  3/48,     and     3/48 :  3  =  1/48, 

.-.  the  arithmetic  series  is     1/12,     1/1 G,     1/24,     1/48, 
and        the  harmonic   series  is         12,         IC,         24,  48. 

THE   ANALOGIES   OF   THE   THREE   PROGRESSIONS. 

Note.  The  analogies  and  relations  of  the  three  progres- 
sions may  be  thus  stated:  if  p,  q,  r  be  three  numbers 

in  arithmetic  progression,  then    p  —  q\q  —  r=p\p\ 

in  geometric  progression,  then    p  —  q-.q  —  r  =p :  q ; 

in  harmonic    progression,  then    p  —  q:q  —  r  =p :  r. 

The  three  means  are     i(p-\-r),     .\/pry     2pr/{p-\-r)\     and 

the  geometric  mean  of  py  r  is  the  geometric  mean  of  their 

arithmetic  and  harmonic  means. 


PR.  3,  §3]  HARMONIC   PROGRESSION.  227 

QUESTIONS. 

Continue  the  harmonic  seiies  for  three  terms  in  each  direction: 
1.  2,  3,  6.         2.  3,  4,  G.         3.  1,  H,  If.        4.   li,  U,,  If. 

5.  The  harmonic  mean  of  two  numbers  is  twice  the  prod- 
uct of  the  numbers  divided  by  their  sum. 

Insert  harmonic  means  as  follows: 

6.  Five  between  1/3  and  1/5. 

7.  Three  between  7/5  and  7/13. 

8.  Five  between  4/5  and  —8/11. 

9.  Three  between     l/{4a  +  b)     and     l/5b. 

10.  AYhatever  be  the  values  of  p,  r,     {p  —  vY    is  positive, 
and    p^\-r'^>2pr',     \(p  +  r)>\/pr,     4p)h''^/{p^-rY<pr. 

11.  Prove  the  analogies  of  the  tliree  progressions  as  above. 

12.  Tlie  geometric  mean  between  two  numbers  is  8,  the 
harmonic  mean  6f :  find  the  numbers. 

13.  The  difference  of  two  numbers  is  8  and  their  harmonic 
mean  is  If:  what  are  the  numbers? 

14.  The  arithmetic,  geometric,  and  harmonic  means  of  two 
numbers  greater  than  unity  are  in  descending  order  of  ning- 
nitude. 

15.  The  arithmetic  mean  between  two  numbers  is  3  and  the 
harmonic  mean  2|:  find  the  numbers. 

16.  If  z  be  the  harmonic  mean  of  a,  h,  then 

\/{z-a)-Vl/{z-h)^\/a  +  l/b,    . 

17.  AVhat  number  must  be  added  to  each  of  three  given 
numbers  a,  b,  c,  that  the  three  results  may  be  in  liarmonic 
progression  ? 

18.  If  a,  b,  c  be  in  harmonic  progression,  tlien 

\/{a  -b)^  \/{b  -  c)  +  \/{c  -  a)  =  \/c  -  \/a. 

19.  If  to  each  of  three  consecutive  terms  of  a  geometric 
progression  the  second  of  the  three  be  added,  the  sums  are  in 
harmonic  progression. 

20.  If  G  be  the  geometric  mean  of  A,  b,  then  the  liarmonic 
means  of  A,  G,  and  g,  b  are  equal. 


228  PROGRESSIONS,  INCOM.  POWERS,  LOGARITHMS.  [Vm,LEM. 

§  4.   INCOMMENSURABLE   POAVERS. 

If  ?'\  r^,  1^,  '  '  •  r"  be  a  series  in  geometric  progression, 
the  exponents  1,  2,  3,  •  •  •  n  form  ji  series  in  arithmetic  pro- 
gression; and  tlie  numbers  1,  2,  3,  •  •  •  n  serve  also  to  define 
the  position  of  the  terms  of  the  geometric  series,  r^  being  the 
first  term,  Ti ,  r^  the  second  term,  Tg,  r^  the  third  term, 
T3 ,  •  •  •  r"    the  nth.  term,  T,i . 

If  two  geometric  means  be  interpolated  between  every  pair 
of  consecutive  terms  of  this  geometric  series,  the  exponents  of 
r  in  the  new  series  form  the  arithmetic  series  1,  1^,  If,  2,  2^, 
2 1,  3,  •  •  • ;  and  here,  too,  the  exponents  serve  to  define  the 
position  of  the  terms  of  the  geometric  series,  the  terms  of  the 
original  series  being  the  major  terms,  or  integer  terms,  and 
the  others  tlie  minor  terms,  or  fraction  terms. 

So,  for  other  means  that  may  be  interpolated  between  pairs 
of  terms,  however  great  their  number;  and  r"  is  the  ^^th  term, 
T„,  whether  ?i  be  an  integer  or  a  fi-action,  positive  or  negative. 
The  exponents  form  an  arithmetic  series  of  commensurable 
numbers,  and  the  corresponding  terms  of  the  geometric  series 
are  commensurable  powers  of  r. 

But  if  the  arithmetic  series  be  made  continuous,  then  also 
the  geometric  series  is  a  continuous  series  of  the  powers  of  r. 
Among  the  terms  of  this  continuous  arithmetic  series  are 
included  incommensurable  numbers,  and  the  corresponding 
terms  of  the  continuous  geometric  series  are  incommensurable 
powers  of  r,  incommoisurable  powers  being  distinguished 
from  commensurable  powers' as  powers  whose  exponents  are 
i  n  CO  m  m  e n  s u  rable  n u  mbers. 

Lemma.  Tlie  ratio  of  two  terms  of  a  geometric  series  is  t/iat 
power  of  til e  common  ratio  icliose  exponent  is  tlie  excess  of  tlie 
number  ivUich  defines  the  posltio7i  of  tlie  first  term  over  that 
which  defines  tlie  other.  [df.  geom.  jn-og. 

E.g.,    T9:T5=rS     T2o:Tia=r%    Ti/2:Tv3  =  ?''^     t„:t^:=  r^-^. 


$•4]  INCOMMENSURABLE   POWERS.  ^.  229 

questto:n-s. 

1.  In  the  geometric  series  •  •  •  1/64,  1/8,  1,  8,  64,  •  •  •  find 
the  common  ratio.     What  is  To?  Tj  ?  t_i?  t.?  t_2?  T3  ?  t_3? 

Interpolate  two  means  between  every  pair  of  consecutive 
terms,  and  name  the  terms  so  interpolated. 

2.  In  the  geometric  series  •  •  •  ,  16/81,  4/9,  1,  9/4,  81/16, 
•  •  •  ,  find  the  common  ratio,  and  name  the  terms. 

Interpolate  three  means  between  every  pair  of  consecutive 
terms,  and  name  the  terms  so  interpolated. 

3.  On  a  horizontal  line  as  axis  take  a  point  o,  and  on  this 
axis  lay  off  distances  to  the  right  and  left  from  0  proportional 
to  the  positive  and  negative  exponents  of  tlie  series  in  ex.  2; 
at  the  points  so  found  draw  vertical  lines  and  lay  off  distances 
upward  proportional  to  the  terms  themselves;  join  the  upper 
ends  of  these  vertical  lines,  in  their  order,  by  straight  lines, 
thus  forming  ^  plat  of  the  serie.<,  that  rises  faster  and  faster. 

4.  If  a  dollar  be  put  at  compound  interest  at  the  annual 
rate  of  10  per  cent,  find  the  amount  at  the  end  of  1  year;  2 
years;  3  years;  •  •  •  8  years.  These  amounts  form  a  true  geo- 
metric series  with  the  common  ratio  1.1. 

5.  In  ex.  4,  the  amount  would  be  the  same  at  the  end  of 
any  period  of  years  if  the  interest  were  compounded  half 
3^early  at  the  ratio  yi.l,  quarterly  at  the  ratio  -v^l.l, 
monthly  at  the  ratio     f^l.l,     and  so  for  any  shorter  periods. 

What  relations  have  the  arithmetic  series  of  exponents,  the 
growing  time,  and  the  geometric  series  of  amounts? 

6.  In  ex.  4,  as  the  time  grows  continuously,  so  may  the  in- 
terest and  the  amount,  i.e.,  just  as  soon  as  any  interest  is 
earned,  that  interest  may  itself  become  principal  and  begin  to 
earn  interest;  and  the  plat  of  the  growing  amount  is  a  con- 
tinuous curve,  rising  faster  and  faster  from  the  axis. 

7.  Growing  continuously,  the  amount  in  ex.  4  becomes 
double  the  principal  at  some  time  between  seven  and  eight 
years.  This  time  is  definite  and  distinct;  it  is  not  an  integer, 
and  not  a  simple  fraction;  hence  it  is  incommensurable,  and 
2  is  an  incommensurable  power  of  1.1. 


230  PROGFFSSIONS,  INCOM.  POWERS,  LOGARITHMS.  LVlII,  Taa. 
THE   PRODUCT   OF   POWERS  OF  THE   SAME   BASE. 

Theor.  6.  The  prodiLct  of  two  or  more  powers  of  a  base  is 
that  power  of  the  base  whose  exponent  is  the  sum  of  the  expon- 
ents  of  the  factors. 

Let  m,  n  be  any  two  numbers,  A  any  base,  then  a*"-a"  =  a"*+". 
(«,  b)  m,  n  both  commensurable :  [VI,  th.  4. 

(c)  m,  n  either  or  both  of  them  incommensurable. 
For,  let  the  geometric  series  a^  a^,  a^  •  •  •  ,  whose  common 
ratio  is  a,  be  made  continuous  by  letting  the  exponent 
grow  continuously  through  all  the  intermediate  values, 
commensurable  and  incommensurable; 
then  *.•  A"*=:T^,    whatever  ?;*  may  be,  and    a'"  +  "  =  t^  +  „,   [df. 
and   •.•T^+„     is  the  ni\\  term  beyond  T„j,  and   its  value  is 
A"* -A",  [lem. 

.-.  A"*-A"  =  A"*+"; 
and  so  for  three  or  more  powers.  Q.E.D. 

A    POWER   OF   A   POWER. 

Theor.  7.  A  poiver  of  a  poiuer  of  a  base  is  that  poiver  of  the 
base  whose  exponent  is  the  product  of  the  given  exjjonents. 
Let  m,  n  be  any  two  numbers,  A  any  base;   then  (a"')"  =  a*""". 
(rt,  b)  m,  n  both  commensurable :  [VI,  th.  2. 

(c)  m,  n  eitlier  or  both  of  them  incommensurable. 
For,  let  the  geometric  series    aS  a^,  a^  •  •  •  ,    whose  common 
ratio   is  A,  be  made  continuous,  and,  in   tliis   series, 
mark    A"*,  (a'")^  (a"*)^  •  •  •     as  the  principal  terms  of 
a  series  whose  common  ratio  is  A"^; 
then  *.*  (a*")**   is  both  the  ni\\  term  of  this  series  and  the  mnt\\ 
term  of  the  original  series,  whose  value  is  a'"  ",  [df 
.-.  (a"')"  =  a'"".  <^.E.I). 


6,T,8,  §4]  IXCOMMEKSURAELE   POWERS.  231 

THE    PKODUCT   OF    LIKE    POWERS   OF    DIFFEKEXT   BASES. 

Theor.  8.    TJie  product  of  like  2J0ivers  of  two  or  )nore  lases 
is  the  like  power  of  their  product. 
Let  n  be  any  number  and     A,  b,  c,  •  •  •     an}'  bases; 
then     A''-B"-c"-  •  •  =(a.b-c-  •  •)"'. 

(<7,  I)  n  commensurable  :  [VI,  tli.  5. 

(c)  n  incommensurahle. 
For,   let  b  =  a'"; 
then  A^ -3"  =  a'^- a'""  [th.  7. 

=  A' 


A  "  +  "»-'! 


[th.  6. 

=  A(^  +  '"^"r=(A-A'")'*  [th.  T. 

=  (a-b)";  [above, 

and  so  for  tlu'ce  or  more  bases.  Q.e.d. 

QUESTTOXS. 

1.  If  the  interest  be  compounded  instantly  at  a  rate  that  is 
equivalent  to  the  annual  rate  10,  and  if  tlie  i)rincipal  be 
doubled  in  m  years,  and  this  double  be  tripled  in  //  years  more, 
the  whole  time  is  m+.n  years,  and  the  final  amount  is  six- 
fold the  first  principal: 

/.^.,  if     2p=p-{l.\y'\     and     Qp  =  2p-{\.iy\ 

then  •.•  6jo=jo.(l.l)'"-(l.l)%     and     G;j  =  />  (1. !)'«+«, 

2.  If  the  interest  be  compounded  instantly  at  a  rate  that  is 
equivalent  to  the  annual  rate  10,  and  if  the  principal  be 
doubled  in  m  years,  and  this  double  be  tripled  in  n  periods  of 
w  years,  the  whole  time  is  m  -n  years,  and  the  final  amount  is 
sixfold  the  first  principal: 

i.e.,\i     2p=p-{\.\Y'      and     G7;=;>-2"; 
then  •.•  6/;=:/)-[(l.l)"']",     and     Q>p=p'{l.iy'-^. 
.-.  [(1.1)"']"  =  (1.1)'"". 


232   PROGRESSIONS,  INCOM.  POWERS,  LOGARITHMS.  LVIII.Ths. 

§5.    LOGAEITHMS. 

The  logarithm  of  a  uiimber  is  the  exponent  of  that  power  to 
which  another  number,  the  base,  must  be  raised  to  give  the 
number  first  named. 

E.g.,  in  the  equation   a^  =  n,   a  is  the  base,  N  is  the  number, 
and  X  is  the  exponent  of  A  and  the  logarithm  of  ]sr. 

Operations  upon  or  with  logarithms  are  therefore  operations 
upon  or  with  the  exponents  of  the  powers  of  the  same  base; 
and  the  principles  established  for  such  powers  apply  directly 
to  logarithms,  with  but  the  change  of  name  noted  above. 

The  three  equations  A'*'"=]sr,  x^logjs,  N  =  logr^^  are 
equivalent  equations.  The  second  is  read,  x  is  the  logarithm 
of  N  to  the  base  A,  or  x  is  the  A-logarithm  o/n;  and  the  last 
means  that  N  is  the  number  whose  logarithm  to  the  base  A 
is  x;  it  is  read,  K  is  the  anti-logarithm  of  x  to  the  base  A. 
E.g.,  O^log^l  and  l  =  logr^O,  whatever  A  may  be. 
So,      l  =  logo2,         2  =  log39,         3=:log4G4,  4  =  log5625, 

and     2  =  log3-%       9  =  log3-%     C4  =  logr^3,       625  =  log,-H, 
So,    -lr=log2l/2,  -2==log3l/9,  -3  =  logJ/G4,     -4  =  log,l/G25, 
and  -l  =  logv22,     -2  =  logi/39,     -^  =  \ogyM,       -4  =  logi/5625. 

If  the  base  bo  well  known  it  may  be  suppressed,  and  these 
two  equations  may  then  be  written     o.'^logN',     N=:log~^r. 

If  while  A  is  constant  N  take  in  succession  all  possible 
values  from  0  to  oo,  the  corresponding  values  of  x  constitute  a 
system-  of  logaritlims  to  the  base  A. 

Theor.  9.   Tlie  logarithm  of  unity  to  any  base  is  zero, 

Theor.  10.   The  logarithm  of  the  base  itself  is  U7iity, 

Theor.  11.  If  the  base  be  positive  and  larger  than  iiiiifg,  the 
logarithms  of  numbers  greater  than  unity  are  ])ositive,  while 
of  numbers  positive  and  smaller  than  unity  they  are  negative; 
and  if  the  base  be  positive  and  smaller  than  tinity,  the  log- 
arithms of  numbers  greater  t]ia)i  unity  are  negative^  while  of 
numbers  positive  and  smaller  than  unity  they  are  positive. 


9, 10, 11,  S  5  3  LOGARiTHMS.  23cJ 

QUESTIONS. 

With  4  as  base,  find  the  logarithms  of: 

1.  16;     8;     1;     64;     32;     256;     4"^;     16-^;     32-^/*. 

2.  .25;     16^2;     1/16;     1/8;     128/1024;     |/2/256. 

3.  Are  the  numbers  below  commensurable  or  incommen- 
surable powers  of  4  ? 

5;    25;    125;    625;    .5;    .25;    .125;    .0625. 
Between  what  commensurable  powers  do  the  incommen' 
surable  powers  lie  ? 

4.  What  is  the  logarithm  of  144  to  the  base  2^/^  ? 

5.  What  effect  is  produced  on  the  logarithm  of  a  number 
by  making  the  base  smaller  ?  by  making  it  lai-ger  ? 

6.  Find  logs  3125;     logr  343"^;     logi/3  81;     logi/7  343. 

7.  Find,  to  the  base  a,     log  ^a-''>/^\     log  [{a-''''^)-^/''\-y\ 

8.  With  9  as  base  find  the  anti-logarithm  of: 

h     -1;     5/2;     -3/2;     2;     "2;     0;     1;     3/4;     "3/4. 

9.  With  8  as  base,  write  a  series  of  six  logarithms. 
What  base  makes: 

10.  log  64  =  2?  log  6i  =  2?   Iog-i3=-1000?   log  125  =  3? 

11.  log-i|  =  343?  log-i-|  =  32?   log2i  =  i?   log64=-3? 

12.  A  number  has  different  logarithms  to  different  bases, 
and  it  may  have  the  same  logarithm  to  two  different  bases, 
but  with  a  given  base,  it  can  have  but  one  logai'ithm. 

13.  With  a  negative  base  what  positive  numbers  and  what 
negative  numbers  have  logarithms?  What  negative  numbers 
have  logarithms  to  positive  bases  ? 

In  making  computations  by  logarithms,  when  may  the  signs 
of  the  numbers  be  disregarded  ? 

14.  With  a  base  smaller  than  unity,  what  is  the  logarithm 
of  a  very  large  number?  of  unity?  of  a  very  small  number? 
of  zero?  of  the  base  itself? 

15.  With  a  positive  base  larger  than  unity,  what  is  the 
logarithm  of  a  very  small  fraction  ?  of  zero?  of  a  very  large 
number?  of  unity?  of  the  base  itself? 


234  PROGRESSIONS,  INCOM.  POWERS,  LOGARITHMS.  [Vlli,  f hs. 
LOGARITHMS   OF    PRODUCTS   AND    QUOTIENTS. 

Theor.  12.  Tlie  logaritlim  of  a  product  is  fl/e  si(7)i  of  the 
logarithms  of  the  factors;  and  the  higaritlDu  of  a  ([uotient  is 
the  excess  of  the  logaritlim  of  tlie  dividend  over  that  of  the 
divisor.  [th.  G. 

E.g.,    logjB-c:D)  =  log,  B  +  log.,c-log,D. 

logarithms    of    powers   and    ROOTS. 

Theor.  13.  The  logaritlim  of  a  jxiwer  of  a  nnmher  is  the 
product  (f  tlie  logarithm  of  tlie  nnmher  hy  tlie  eocponent  of  the 
power  sought;  that  of  a  root  is  the  quotient  of  the  logarithm 
hy  the  root-index.  [th.  7. 

^.g.,   logaB^.>^c)  =  2log.B  +  ilog,c. 

CHANGE    OF    BASE. 

Theor.  11.  If  the  logarithm  of  a  nnmler  he  taken  to  two 
differe7it  bases,  the  first  logarithm  is  the  jyroduct  of  the  second 
logarithm  into  the  logarithm  of  the  second  base  taken  to  the 
first  base,  and  vice  versa. 

For,  let  N  be  any  luimber.  A,  B  two  bases,  and     yz=i\og^^\ 
then  •.•  N  =  B^,  [df.  log. 

.-.  log,  N  =  log^  B^  =  ?/ .  log^  B,  [th.  1 3. 

=  log3N-log^B.  •         Q.E.D. 

So,  logB  N  =  log^  N  •  l0g3  A.  Q.  E  D. 

Cor.  1.   The  logarithms  of  two  numbers,  each  taken  to  the 
other  number  as  base,  are  reciprocals. 
For,  let  B  =  A*, 
then  ♦.•   A  =  B^/^  log^  B  =  .r,  log^  A  =  l/x,  [d f.  log. 

.-.  log^  B  •  logn  A  =  .r  •  \/x  =1.  Q.  E.  D. 

Note.  There  are  two  systems  of  logarithms  in  use: 
natural  logarithms,  whose  base  is  e  [2.71828- •  •],  and  com- 
mon logarithms,  whose  base  is  10.  Their  relations  to  each 
other  are  expressed  by  the  equations 

logioN  =  log.N  •  logjo'-,     logio^'  = .  •424294. 


12, 13, 14.  §5]  LOGARITHMS  S36 

QUESTIONS. 

If  the  logarithms  of  .r,  ?/,  r/,  h  be  kiiown^  show  how  to  find: 
1.  log  (|/.i-^-|///).        2-  log  (|/rt-'^-|/«Z»-3).       3.  log  abxy. 
4.  log(^/Z^:.T^)l  5.1og(r^r:^*^)-i'l  0.  log(^/i/V>-^/3^-i/*:?//5). 

Given     log^o  2  =  .3010,     log^o  3  =  .-4rri,     log^o  7  =  .8451,    find: 


7. 

logs. 

8. 

log  0. 

9. 

log  8. 

10.  log  9. 

11. 

log  10. 

12. 

log  12. 

13. 

log  14. 

14.  log  15. 

15. 

log  IG. 

10. 

log  20. 

17. 

log  18. 

18.  log  21. 

19. 

log  24. 

20. 

log  25. 

21. 

log  28. 

22.  log  30. 

23. 

log  |/72. 

24. 

log  300^/1 

25. 

log  IGl 

2G.  log  1728. 

27. 

log  2/5. 

28. 

log  3^. 

29. 

log  Ah. 

30.  log  12i. 

31. 

log  If. 

32. 

log  ^|/15. 

33. 

log  ^(3-^ 

.5^:|/2). 

34. 

log  ^(729 

.^9- 

-i.27-*/3\ 

35. 

log  (27- 

13.^4-1/6)^ 

3G.  What  is  the  logarithm  of  the  arithmetic  mean  of  15,  21  ? 
of  the  harmonic  mean?  of  the  geometric  mean  ? 

37.  The  logarithms  of  two  given  nnmbers  bear  a  constant 
ratio  to  each  other,  whatever  the  base. 

38.  Given    log7  49  =  2,    and    logio  7  =  .8451,    find   log^o  49. 

39.  Given    logi6G4  =  3/2,     log^o  IG  =  1.2040,    find    log^o  G4. 

40.  If     log:?-y  =  r/,     log.T/rt=:^,     find    log  .t    and    log  i/. 
From  the  logarithms  of  2,  3,  5,  7  to  the  base  10,  above,  find : 

41.  log7  10.  42.  log5  10.  43.  \og.,  10.  44.  log.  10. 
45.  log;  700.  4G.  log5  7.  47.  logs  4/30.  48.  log,  1^. 
49.  logs  2.  50.  Iog2  800.  51.  log,  3=l.  52.  logs  270. 
53.  log2  5.  54.  logs  343.  55.  logg  28.  56.  log,  14f 

57.  log  75/1G-2  log  5/9  +  log  32/243  =  log  2. 
From    logio2  =  . 30103    and     logio  7  =  .84509,    find: 

58.  log;  |/2.         59.  logt/2  ^.         GO.  log,  4/7.       Gl.  log^^  2. 


236  PROGRESSIONS,    INCOM.  POWERS,  LOGARITHMS.  [VIU.Th. 
SPECIAL    PROPERTIES   OF   THE    BASE    10. 

The  logarithm  of  au  integer  power  of  10  is  an  integer. 
E.g.,    of     1000,     100,     10,     1,     .1,     .01,     .001, 
the  logarithms  to  the  base  10  are 

+  3,       -2,     n,     0,    -1,      -2,        -3. 
But  of  any  other  number  the  logaritlim  is  fractional  or  in- 
commensurable;   and  if  incommensurable,   it  consists  of   a 
whole  number,  the  characteristic,  and  an  endless  decimal,  the 
mantissa. 

As  a  matter  of  convenience  the  mantissa  is  always  taken 
positive;  and  the  characteristic  is  the  exponent,  positive  or 
negative,  of  that  integer  power  of  10  which  lies  next  below 
the  given  number. 

A  negative  characteristic  is  indicated  by  the  sign  -  above  it. 
E.g.,  the  logarithms  of  the  numbers 

2000,  20,  .2,  .002, 

are       3.30103...,     1.30103..-,     1.30103...,     3~30103..., 
whose  characteristics  are     3,     1,     1,     3,   and  whose  common 
mantissa  is  .30103.  .  .. 

Theor.  15.  If  a  number  he  multiplied  {or  divided)  by  anj/ 
integer  power  of  10,  ilie  logarithm  of  the  product  (or  quotient) 
and  the  logarithm  of  the  number  have  the  same  mantissa. 
For  •.*  the  logarithm  of  a  product  is  the  sum  of  the  logarithms 
of  its  factors,  [th.  12. 

and  •.•  the  logarithm  of  the  multiplier  is  an  integer,        [hyp. 
/.  the  mantissa  of  the  sum  is  identical  Avith  tlie  mantissa 
of  the  logarithm  of  the  multiplicand.         q.e.d. 
So,  if  a  number  be  divided  by  an  integer  power  of  10. 

Cor.    The  logarithms  of  all  numbers  expressed  by  the  same 
figures  in  the  same  order  have  the  same  mantissa,  but  different 
characteristics. 
E.g.,  the  logarithms,  correct  to  four  figures,  of  the  numbers 

70500,      795,  7.95,         .0795,       .000795, 

are       4.9004,     2.9004,     0.9004,     2.9004,     4.9004. 


15.  §5]  LOGARITHMS.  237 

QUESTIONS. 

1.  What  kind  of  power  of  10  is  a  number  whose  logarithm 
is  a  simple  fraction  ? 

Can  such  a  power  of  10  be  a  commensurable  number  ? 
Conversely,  what  kind  of  logarithm  has  a  commensurable 
number  that  is  not  an  integer  power  of  10  ? 
What  is  the  mantissa  of  such  a  logarithm? 

2.  What  kind  of  series  is  formed  by  the  powers  of  10  on 
the  opposite  page  ?  what  by  their  logarithms? 

So,  with  the  series  2000,  20,  .2,  .002  and  tliuir  logiirithms: 

3.  Xame  some  base  that  gives  a  rising  series  of  logarithms 
for  a  falling  series  of  numbers. 

4.  What  relation  has  the  difference  in  the  series  of  log- 
arithms to  the  ratio  in  the  corresponding  series  of  numbers  ? 

5.  Explain  why  the   logarithms   of     9520,     95.2,     .952, 
.00952,     have  the  same  mantissa. 

What  are  the  characteristics  of  these  logarithms? 

6.  Moving  the  decimal  point  one  place  to  the  left  in  a 
number  has  what  effect  on  the  characteristic  of  its  logarithm? 

So,  moving  the  point  three  j^laces  to  the  right  ?  • 

Given     log  4096000  =  6.6124,     find : 

r.  log  4096.      8.  log  40.96.      9.  log  6.4.         10.  log  8. 
11.  log  4.  12.  log  512.       13.  log  .016.       14.  log  .0002. 

15.  How  many  figures  in  10^  ?  in  any  number  between  10* 
and  10^?  in  a  number  between  10" "^  and  10"? 

What  is  the  characteristic  of  the  logarithm  of  an  integer 
expressed  by  three  figures?  by  n  figures?  How  many  figures 
are  there  in  the  anti-logarithm  if  the  characteristic  be  5  ? 

16.  Given   log  2  :=:  .30103,   how  many  figures  are  there  in  2^"? 

17.  The  logarithm  of  a  decimal  fraction  has  a  negative 
characteristic,  a  unit  larger  than  the  number  of  ciphers  that 
follow  the  decimal  point. 

18.  If  (1/2)^*^  be  reduced  to  a  decimal,  how  many  ciphers 
follow  the  decimal  point  ? 


2'SS    PROGRESSIONS,  INCOM.  POWERS,  LOGARITHMS.  [vm,Pa 
TABLES   OF   LOGARITHMS. 

The  logarithms  of  any  set  of  consecutive  numbers^  arranged 
in  a  form  convenient  for  use,  constitute  a  table  of  logdvitknis. 
A  table  to  the  base  10  need  give  only  the  mantissas;  the  cliar- 
acteristics  are  evident.  The  difference  of  two  consecutive 
mantissas  is  tlieir  tabular  difference. 

Tables  may  be  carried  to  any  number  of  decimal  places; 
but  the  four-place  table  on  pages  244,  245  is  sufficiently  accu- 
rate for  ordinary  use.  The  fourth  figure  is  in  error  by  less 
than  half  a  unit.  The  first  two  figures  of  each  number  are 
printed  at  the  left  of  the  page,  and  the  third  figure  at  the  top 
of  the  page,  over  the  mantissa  of  the  corresponding  logarithm. 

Pros.  4.  To  take  out  the  logarithm  of  a.  number. 
(a)  A  three-figure  number :  take  otit  the  tabular  mantissa  that 

lies  in  line  tvith  the  first  twofigicres  and  under  the  third; 
the  characteristic  is  the  exponent  of  that  integer  power  of  10 

which  lies  next  beloiu  the  given  number. 
If  a  number  have  one  or  two  figures,  make  it  three-figured  by 

annexing  zeros. 
E.g.,    log  567=: 2. 7536;     log  5.6  =  0.7482;     log  .05  =  2.6090. 
{b)  A  number  of  more  than  tliree  fiqures:  take  out  the  tabuhtr 

mantissa  of  the  first  three  figures; 
subtract  this  mantissa  from  the  next  greater  tabular  mantissa; 
multijyly  the  difference  so  found  by  the  remaining  figures  <f 

the  immber  as  a  decimal; 
add  this  product,  as  a  correction,  to  the  mantissas  of  the  first 

three  figures. 
E.g.,  to  find  log  500.0: 

then  vlog  500  =  2.0990,     log  501  =  2.0998,  [tables, 

and       log  501 -log  500  =  .0008,     500.0 -500  =  .6, 

.-.log  500.0  =  2.0990  +  6  tenths  of  .0008  =  2.0995. 
The  rule  for  interjiolation  rests  upon  this  property  of  log- 
arithms, that  their  differences  are  nearly  proportional  to  the 
differences  of  the  numbers  when  tlie  differences  are  small. 


4.  §5)  LOGARITHMS.  239 

QUESTIONS. 

1.  How  docs  a  table  of  logarithms  of  prime  numbers  make 
it  possible  to  find  the  logaritlims  of  all  other  nnmbeis ? 

2.  Why  do  not  tables  of  common  logarithms  contain  char- 
acteristics ?  must  they  be  given  in  tables  to  other  bases  ? 

3.  Find  log  2  -  log  1000,  giving  a  wholly  negative 
logarithm  as  the  result,  and  show  that  it  is  the  same  as 
log  1/1000  +  log  2,     or     3. 30103. 

4.  In  getting  logarithms  from  a  table,  by  what  right  and 
for  what  purpose  are  zeros  annexed  to  numbers  having  fewer 
than  three  significant  figures? 

From  the  table  take  out  the  logarithms  of: 

5.  12.  6.  120.  7.  123.  8.   124. 

9.   123.4.  10.  1.234.  11.  12350.  12.   .001235. 

13.  In  finding  log  73265,  to  how  large  a  difference  in  the 
number  does  the  tabular  difference  correspond  ? 

AVhat  part  of  this  difference  is  the  rest  of  the  number? 
Take  out  the  logarithms  of: 

14.  9032.  15.  .00064.  16.  75.15.  17.  6.872. 
18.  .25.             19.  2496000.        20.  .00854.         21.  1000000. 
22.  24G.3.         23.  .9467.             24.  .007009.       25.  1463. 

26.  Show  that  the  last  ^laragraph  in  case  (a)  of  prob.  4 
might  read:  ^nal'e  tlie  cliaracteristic  one  less  than  the  mimher 
ofjinures  in  the  integer  imrt  of  the  nnmber. 

By  use  of  the  table  on  pp.  244,  245,  find  the  logarithm  of: 

27.  43962.  28.  521.6701.  29.  .004281.  30.  124365000. 
31.  2.76314.  32.  .4580012.  33.  6309.25.  34.  .00.0519328. 
35.  Show  how  the  logarithm  of  a  number  lying  between  two 

tabular  numbers  mny  be  found  from  the  laiger  of  the  two 
tabular  logarithms. 

30.  Find  in  the  table  log  60,  log  60.5,  log  61,  and  show  that, 
while  the  second  of  these  logarithms  is  almost  half-way  between 
the  other  two,  in  numbers  differing  more  widely,  e.fj.^  60^  70, 
i80,  this  proportion  does  not  hold  truCj 


240    PROGRESSIONS,  INCOM.  POWERS,  LOGARITHMS.   [vm.Pa. 
PrOB.   5.    To    TAKE    OUT   A   NUMBER   FROM   ITS    LOGARITHM. 

(a)  The  mantissa  found  in  the  table:  join  tlie  figure  at  the  top 

that  lies  above  the  given  mantissa  to  the  tiuo  figures 
upon  the  same  line  at  the  extreme  left; 
in  the  tliree-figure  number  thus  found  so  place  the  decimal 
point  that  the  number  shall  lie  next  above  that  power  of 
10  whose  exponent  is  the  given  characteristic. 

E.g.,  log-i  2.7536:=  567;  log"^  0.7483  =  5.6;  log-^  2.6990  =  .05. 

(b)  The  mantissa  not  found  in  the  table:  take  out  the  three- 

figure  anti-logarithm  of  the  tabular  mantissa  next  less 
than  the  given  mantissa; 
and  to  it  join  the  quotient  of  the  difference  of  these  tiuo  7nan- 
tissas  by  the  tabular  difference. 

E.g.,  to  take  out  log-^  2.6995: 

then  •.*  the  next  less  tabular  mantissa  is  .6990,  and  the  next 
greater  .6998, 
.'.  the  tabular  difference  is  .0008; 
i.e.f  an  increase  of  .0008  in  the  mantissa  .6990  corresponds  to 

an  increase  of  1  in  the  number. 
But   *.•  the  given  mantissa  differs  from  .6990  by  .0005, 

/.  this  difference  corresponds  to  an  increase  in  the  anti- 
logarithm  of  .0005/.0008,  i.e.,  of  .6. 
and    -.-log-i  2.6990  =  500, 

.'.  the  figures  in  the  number  sought  are  5006,  and  the 
characteristic  2  shows  that  the  number  is  5U0.6. 

So,  to  find  log-*  3.4986: 

then  •.*  the  given  mantissa  lies  between  .4983  and  .4997, 
and        .4986 -.4983  =  .0003,         .4997 -.4983  =  .0014, 
.0003/.  0014  =  .2, 
.-.  log-*  .4986  has  the  figures  3152, 
and    log-*  3.4986  =  .003152. 

The  process  is  but  the  inverse  of  that  for  taking  out  log- 
arithms, and  the  reason  of  the  rule  is  the  same  for  both. 


5,  §5]  LOGARITHMS.  241 

LIMITATIONS    IN   THE    USE    OF    THE    TABLES. 

The  possible  erroi-  of  any  logarithm,  as  printed  in  this  table, 
is  half  a  unit  in  the  fourth  place,  and  the  possible  error  of 
any  tabular  difference  is  a  unit;  but  the  probable  error  is 
much  less.  The  fourth  figure  of  the  anti-logarithm,  the  first 
got  by  division,  is  generally  trustworthy;  the  fifth  is  rarely  to 
be  used.  The  possible  error  in  the  result  is  nearly  ten  times 
greater  if  the  logarithm  be  near  the  end  of  the  table  than  if 
near  the  beginning,  for  then  the  tabular  difference,  the  divisor, 
is  much  smaller,  and  an  error  either  in  it  or  in  the  dividend 
has  greater  effect.  If  greater  accuracy  be  desired,  larger 
tables  must  be  used. 

QUESTIONS. 

From  the  table  take  out  the  anti-logarithms  of: 

I.  1.0792.  2.  2.0792.  3.  2.0899.  4.  2.0934. 
5.  2.0913.             6.  0.0913.             7.  4.0917.  8.  1.9652. 

9.  Show  that  the  last  paragraph  of  case  (a)  in  prob.  5  might 
rend:  make  the  number  of  figures  in  ilie  integer  part  of  the 
number  one  more  than  the  characteristic  of  the  logarithm. 

10.  After  dividing  the  difference  between  a  given  mantissa 
and  the  next  less  tabular  mantissa  by  the  tabular  difference, 
why  is  the  quotient  annexed  instead  of  added  to  the  figures 
given  by  the  table? 

Find  the  anti-logarithms  of: 

II.  2.9053.        .  12.  1.7126.  13.  .3402.  14.  1.4612. 
15.  3.0024.          16.  2.5832.  17.  3.7368.  18.  .5505. 
19.  2.5337.          20.  2.6193.  21.  1.8000.  22.  .0971. 

23.  If  the  logarithms  of  a  series  of  multipliers  and  divisors 
be  added,  what  logarithms  must  be  regarded  as  negative? 

If  the  sum  of  any  column  be  negative  it  may  be  made  posi- 
tive by  adding  one  or  more  tens  to  it,  and  subtracting  the  same 
number  of  units  from  the  sum  of  the  next  column  at  the  left. 

24.  Write  under  each  other  the  logarithms  of  3426,  4.003, 
.00162,  324.5,  1.64,  and  then  so  add  them  as  to  find  the  log- 
arithm of     3426  X  4.003  x  ,00162  :  324.5 ;  1.64. 


242  PROGRESSIONS,  INCOM.  POWERS,  LOGARITHMS.  [Mil,  Prs. 

PrOB.  6.    To  DIVIDE  A  LOGARITHM  WHOSE  CHARACTERISTIC 
IS   NEGATIVE. 

W7nte  down  as  the  characteristic  of  the  quotierit  lh^~?tumIier_of 
times  the  divisor  is  contained  in  that  negative  midtiple 
of  itself  tohich  is  equal  to,  or  next  larger  tha/i,  the  nega- 
tive characteristic; 

carry  the  positive  remainder  to  the  niantissa  and  divide. 

E.g.,    4.1234:  3  =  (-G  +  2.1234):  3  =  2.70:8. 

So,        3.4770- 3/2::=  8.43 10/2  =4.2155. 

PrOB.  7.    To  COMPUTE  BY  LOGARITHMS  THE  PRODUCTS,  QUO- 
TIENTS,  POWERS,    AND    ROOTS   OF    NUMBERS. 

For  a  product:  add  the  logarithms  of  the  factors,  and  take  out 
the  anti-logaritliin  of  the  sum. 

For  a  quotient:  from  the  logarithm  of  the  dividend  subtract 
that  of  the  divisor,  and  take  out  the  anti-logarithm. 

For  a  potoer :  multiply  the  logarithm  of  tlie  base  by  the  expo- 
nent ofthepoiver  sought,  and  take  out  the  anti-logarithm. 

For  a  root:  divide  the  logarithm  of  the  base  by  the  root-index, 
and  take  out  the  anti-logarithm. 

E.g.,  to  find  tlie  value  of     (.01519x6.318:7.254)=^/^: 

NUMBERS.  LOGARITHMS. 

.01519  2.1815 

xG.318  +0.8000 

:  7.254  -0.8605 

2.1216x3/2 
3.1824 
and  the  number  sought  is  0.001522. 

PrOB.  8.    To    SOLVE   THE    EXI'ONKNTIAL    EQUATION    A'"  =  B. 

Divide  the  logarithm  ofn  by  the  logarithm  of  the  base  A. 

The  quotient  is  x,  the  exponent  sought. 
For  •.•  A"'=B, 

.-.  .7:-log  A  =  log  B,  wliatever  system  of  logarithms  be  used, 
and       X  =  log  B/log  A.  Q.  e.  d. 


6, 7, 8,  §5]  LO(iAKITHMS.  243 

QUESTIOXS. 

By  logarithms,  find  the  value  of: 

1.  5:5.25  X  1.06-^  2.  575.25x1.03*^. 

3.  -^.00010098.  4.  [TG-^'2.45^/*]2/^ 

^    2^ •5^- 85-^      ^    V(9r-_9^)       ^    V12-.^G5      ^    ^83.64 
3^.;=^    •       '    81.1^5:2*      ''  ^5'i/.lS'       '.081452" 

0.  y. 0000000037591.  10.   i/236.140- y^9G:215. 

11.  What  power  is  2  of  1.05?  3  of  1.04?  4  of  1.03?  5  of  1.02? 

12.  Given  the  logarithms  of  a,  b,  c,  d,  show  how  to  find 
the  value  of     (- a)"-- (-B)y-4c/-D. 

13.  Divide  the  logarithm  3.2G14  by  2,  by  5,  and  by  G,  in  turn. 

14.  Find  the  cube  root  of  log~^  7.3550. 

15.  Find  the  cube  of  the  fourth  root  of  log-^  6.5448. 
Find  by  logarithms: 

IG.   The  simple  interest  of  S23.65  for  25  yr.  3  mo.  at  7^  ^. 

17.  The  amount  of  S24G  for  12^  years  if  the  interest  be 
compounded  annually  at  8  per  cent;  if  half-yearly  at  4  per 
cent;  if  quarterly  at  2  per  cent. 

18.  The  20th  term  of  a  geometric  series  if     a  —  b,     r=l}. 

19.  If  the  number  of  births  each  year  be  one  in  forty-five 
of  the  population,  and  of  deaths  one  in  sixty,  in  how  many 
years  will  the  population  double,  taking  no  account  of  other 
sources  of  increase  or  decrease?  triple  ?  quadruple  ? 

20.  From  the  formula  for  compound  interest,  rf=p-  (1  +  r)^, 
find  an  expression  for  t  in  terms  of  r/,  p,  r. 

21.  How  long  must  §1000  be  at  compound  interest  to 
amount  to  S1191.02  at  6  per  cent  a  year?  at  3  percent  half- 
yearly  ?  at  1^  per  cent  quarterly  ? 

22.  In  a  geometric  series,  given     /  =  r7r"~^,     then 

w-l  =  log,.//^,      /i  =  l  +  log,./-log^<^t^l  +  (log  /-log  rO:log  r. 

23.  In  a  geometric  series,  rt  =  5,/ =1280/6561,  r  =  2/3:  find  w. 
J?4.  Solve  the  equations     (5|)'=12i4;     a''^'^/b''-^  =  c'^. 


244 


INCOMMENSURABLE  POWERS,   LOGARITHMS. 


\yui, 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dif. 

lO 

0000 

0043 

0086 

0128 

0170 

0212 

0253 

0294 

0334 

0374 

42 

11 

0414 

0453 

0492 

0531 

0569 

0607 

0645 

0682 

0719 

0755 

38 

12 

0792 

0828 

0864 

0899 

0934 

0969 

1004 

1038 

1072 

1106 

35 

13 

1139 

1173 

1206 

1239 

1271 

1303 

1335 

1367 

1399 

1430 

32 

14 

1461 

1492 

1523 

1553 

1584 

1614 

1644 

1673 

1703 

1732 

30 

15 

1761 

1790 

1818 

1847 

1875 

1903 

1931 

1959 

1987 

2014 

28 

16 

2041 

2068 

2095 

2122 

2148 

2175 

2201 

2227 

2253 

2279 

26 

17 

2304 

2330 

2355 

2380 

2405 

2430 

2455 

2480 

2504 

2529 

25 

18 

2553 

2577 

2601 

2625 

2648 

2672 

2695 

2718 

2742 

2765 

^ 

19 

2788 

2810 

2833 

2856 

2878 

2900 

2923 

2945 

2967 

2989 

22 

20 

3010 

3032 

3054 

3075 

3096 

3118 

3139 

3160 

3181 

3201 

21 

21 

3222 

3M3 

3263 

3284 

3304 

3324 

3345 

3365 

3385 

3404 

20 

22 

3424 

3444 

3464 

3483 

3502 

3522 

3541 

3560 

3579 

3598 

19 

23 

3617 

3636 

3655 

3674 

3692 

3711 

3729 

3747 

3766 

3784 

19 

21 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

3945 

3962 

18 

25 

3979 

3997 

4014 

4031 

4048 

4065 

4082 

4099 

4116 

4133 

17 

26 

4150 

4166 

4183 

4200 

4216 

4232 

4249 

4265 

4281 

4298 

16 

27 

4314 

4330 

4346 

4362 

4378 

4393 

4409 

4425 

4440 

4456 

16 

28 

4472 

4487 

4502 

4518 

4533 

4548 

4564 

4579 

4594 

4609 

15 

29 

4624 

4639 

4654 

4669 

4683 

4698 

4713 

4728 

4742 

4757 

15 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

14 

31 

4914 

4928 

4942 

4955 

4969 

4983 

4997 

5011 

5024 

5038 

14 

32 

5051 

5065 

5079 

5092 

5105 

5119 

5132 

5145 

5159 

5172 

13 

33 

5185 

5198 

5211 

5224 

5237 

5250 

5263 

5276 

5289 

5302 

13 

34 

5315 

5328 

5340 

5353 

5366 

5378 

5391 

5403 

5416 

5428 

13 

35 

5441 

5453 

5465 

5478 

5490 

5502 

5514 

5527 

5539 

5551 

12 

36 

5563 

5575 

5587 

5599 

561 1 

5623 

5635 

5(>47 

5658 

5670 

12 

37 

5682 

5694 

5705 

5717 

5729 

5740 

5752 

5763 

5775 

5786 

12 

38 

5798 

5809 

5821 

5832 

5843 

5855 

5866 

5877 

5888 

5899 

11 

39 

5911 

5922 

5933 

5944 

5955 

5966 

5977 

5988 

5999 

(iOlO 

11 

40 

6021 

6031 

6042 

6053 

60()4 

6075 

6085 

6096 

6107 

6117 

11 

41 

6128 

6138 

6149 

6160 

6170 

()180 

6191 

6201 

6212 

6222 

10 

42 

()232 

6243 

()253 

6263 

6274 

6284 

6294 

6;«)4 

6314 

6325 

10 

43 

6335 

()345 

6:555 

()3()5 

()375 

(;3S5 

6395 

6405 

6415 

6425 

10 

44 

6435 

6444 

6454 

6464 

6474 

6484 

6493 

6503 

6513 

6522 

10 

45 

6532 

6542 

6551 

6561 

6571 

6580 

6590 

6599 

6609 

6618 

10 

46 

6628 

6637 

6(')46 

6()56 

6665 

6675 

6684 

(i<)93 

6702 

<'.712 

9 

47 

6721 

6730 

(5739 

6749 

6758 

67(57 

()77(') 

6785 

67t)4 

(>S()H 

9 

48 

6812 

6821 

6830 

6839 

6848 

6857 

iiSiWi 

6875 

6884 

6803 

9 

49 

6902 

6911 

6920 

6928 

6937 

6946 

6955 

6964 

6972 

6981 

9 

50 

6990 

6998 

7007 

70 1() 

7024 

7033 

7042 

7050 

7059 

70()7 

9 

51 

7076 

7084 

7093 

7101 

7110 

7118 

7126 

7135 

7143 

7152 

9 

r,2 

7160 

7168 

7177 

71.S5 

7193 

7'202 

7210 

7218 

7226 

7235 

8 

53 

71343 

7251 

7259 

7267 

7275 

7'2S4 

7292 

73(M) 

7308 

731() 

8 

54 

73;i4 

7332 

7340 

7348 

7356 

7:}C4 

7372 

73S() 

7388 

739(> 

8 

LOrtARITHMS. 


245 


N 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

Dif. 

55 

7404 

7412 

7419 

7427 

7435 

7443 

7451 

7459 

7466 

7474 

8 

56 

7482 

7490 

7497 

7505 

7513 

7520 

7528 

7536 

7543 

7551 

8 

57 

7559 

7566 

7574 

7582 

7589 

7597 

7004 

7612 

7619 

7027 

8 

58 

76:34 

7&42 

7(>49 

7057 

7004 

7072 

7079 

7686 

7694 

7701 

7 

59 

7709 

7716 

7723 

7731 

7738 

7745 

7752 

7760 

7767 

7774 

7 

60 

7782 

7789 

7796 

7803 

7810 

7818 

7825 

7832 

7839 

7846 

- 

61 

7853 

7860 

7868 

7875 

7882 

7889 

7890 

7903 

7910 

7917 

7 

62 

7924 

7931 

7938 

7945 

7952 

7959 

7900 

7973 

7980 

7987 

7 

63 

7993 

8000 

8007 

8014 

8021 

8028 

8035 

8041 

8048 

8055 

7 

U 

8062 

8069 

8075 

8082 

8089 

8090 

8102 

8109 

8116 

8122 

7 

65 

8129 

8136 

8142 

8149 

8150 

8162 

8109 

8176 

8182 

8189 

~ 

60 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8'^8 

8254 

7 

67 

8261 

8267 

8274 

8280 

8287 

8293 

8299 

8306 

8312 

8:319 

0 

68 

8:325 

8331 

8338 

8a44 

8351 

8357 

8:303 

8370 

8376 

8382 

6 

69 

8388 

8395 

M)l 

8407 

&414 

8420 

8426 

8432 

8439 

8445 

6 

70 

8451 

8457 

&403 

8470 

8470 

8482 

8488 

8494 

8500 

8506 

0 

71 

8513 

8519 

8525 

8531 

8537 

8543 

8549 

8555 

8561 

8567 

0 

72 

8573 

8579 

8585 

8591 

8597 

8603 

8009 

8615 

8621 

8027 

0 

73 

8633 

8639 

8045 

8051 

8057 

866:3 

8009 

8675 

8681 

8080 

6 

74 

8692 

8098 

8704 

8710 

8710 

8722 

8727 

8733 

8739 

8745 

6 

75 

8751 

8756 

8702 

8708 

8774 

8779 

8785 

8791 

8797 

8802 

6 

70 

8808 

8814 

8820 

8825 

8831 

8837 

8*42 

8848 

8854 

8859 

6 

77 

,S8a5 

8871 

8870 

8882 

8887 

8893 

8899 

8904 

8910 

8915 

6 

78 

8921 

8927 

8932 

8938 

8943 

8949 

8954 

8960 

8965 

8971 

6 

79 

8976 

8982 

8987 

8993 

8998 

9004 

9009 

9015 

9020 

9025 

5 

80 

9031 

9036 

9042 

9047 

9053 

9058 

9063 

9069 

9074 

9079 

5 

81 

9085 

9090 

9090 

9101 

9106 

9112 

9117 

9122 

9128 

9133 

5 

82 

9138 

9143 

9149 

9154 

9159 

9105 

9170 

9175 

9180 

9186 

5 

83 

9191 

9196 

9201 

9206 

9212 

9217 

9222 

9227 

9232 

9238 

5 

W 

9243 

9248 

9253 

9258 

9263 

9269 

9274 

9279 

9284 

9289 

5 

85 

9294 

9299 

9:304 

9309 

9315 

9320 

9325 

9330 

9335 

9340 

5 

8<j 

9315 

9350 

9355 

9300 

9:365 

9370 

9375 

9380 

9:385 

9390 

5 

87 

9395 

94(J0 

9405 

9410 

9415 

9420 

9425 

9430 

9435 

9440 

5 

88 

9445 

9450 

9455 

9400 

9405 

9409 

9474 

9479 

9484 

9489 

5 

89 

9494 

9499 

9504 

9509 

9513 

9518 

9523 

9528 

9533 

9538 

5 

90 

9542 

9547 

9552 

9557 

9502 

9500 

9571 

9576 

9581 

9586 

5 

91 

9590 

9595 

9000 

9005 

9009 

9014 

9019 

9624 

962.S 

9633 

5 

92 

mm 

9(>13 

9f'A7 

9652 

9057 

9001 

9066 

9071 

9675 

9680 

5 

93 

9685 

9689 

9694 

9699 

9703 

9708 

9713 

9717 

9722 

9727 

5 

94 

9731 

9736 

9741 

9745 

9750 

9754 

9759 

9763 

9768 

9773 

5 

95 

9777 

9782 

9780 

9791 

9795 

9800 

9805 

9809 

9814 

9818 

5 

90 

9823 

9827 

9832 

9830 

9841 

9845 

9850 

9854 

9859 

9803 

4 

97 

9868 

9872 

9877 

9SS1 

9880 

9890 

9894 

9899 

9903 

99U8 

4 

98 

9912 

9917 

9921 

9920 

99:30 

99:34 

9939 

9943 

9948 

9952 

4 

99 

9956 

9961 

9905 

9909 

9974 

9978 

9983 

9987 

9991 

9990 

4 

246     PROGRESSIONS,  INCOM.  POWERS,  LOGARITHMS.      [Vm, 

§  6.  QUESTIONS   FOR   REVIEW. 
Define  and  iHnstrate: 

1.  A   series;    an  arithmetic  series;    a  geometric  series;  a 
harmonic  series. 

2.  A  rising  series;  a  falling  series  ;  a  continuous  series. 

3.  The  five  elements  of  an  arithmetic  series  ;  of  a  geometric 
series;  the  four  elements  of  a  harmonic  series. 

4.  Arithmetic  means  ;  geometric  means;  harmonic  means. 

5.  The  major  terms  of  a  series;  the  minor  terms. 

G.  A  commensurable  power  ;  an  incommensurable  power. 

7.  A  base  ;  a  logarithm  ;  an  anti-logarithm. 

8.  A  system  of  logarithms  ;  a  table  of  logarithms. 

9.  The  characteristic  of  a  logarithm;  the  mantissa. 

10.  Natural  logarithms;  common  logarithms. 

11.  Tabular   numbers;    tabular   logarithms;    tabular    dif- 
ferences. 


Write  and  prove  the  formulae  for: 

12.  The  last  term  of  an  arithmetic  series;  the  sum. 

13.  The  last  term  of  a  geometric  series;  the  sum. 

14.  The  last  term  of  a  harmonic  series. 

15  The  sum  of  an  infinite  falling  geometric  series;  the 
value  of  a  repeating  decimal. 

IG.  The  arithmetic  mean  of  two  numbers;  the  geouietric 
mean;  the  harmonic  mean. 

State  the  analogies  of  the  three  progressions. 

17.  The  difference  of  any  two  terms  of  an  arithmetic  series. 

18.  The  ratio  of  any  two  terms  of  a  geometric  series. 
Give  a  general  rule,  with  reasons  and  illustrations,  for  : 

19.  Finding  the  other  two  elements  of  an  arithmetic  series, 
when  any  threi^  of  them  are  given;  of  a  geometric  series. 

20.  Inserting  means  in  an  aritiimetic  series;  in  a  geometric 
series  :  in  a  harmonic  series. 


§6.]  QUESTIONS  FOR  REVIEW.  247 

State  the  ^^li'iciple,  with  proof,  that  relates  to: 

21.  The  prodnct  of  incommensurable  powers  oi*  the  same 
base;  the  quotient  of  two  such  powers. 

22.  An  incommensurable  power  of  an  incommensurable 
power  of  a  base. 

23.  The  product  of  like  incommensurable  powers  of  differ- 
ent bases ;  the  quotient  of  two  such  powers. 

24.  What  is  the  logarithm  of  unity  to  any  base?  of  the 
base  itself  ?  of  zero  ? 

25.  If  tlie  base  be  positive  and  larger  than  unity  what  is 
the  logarithm  of  a  number  smaller  than  unity?  of  one  larger 
than  unity?  How  does  the  logarithm  change  as  the  number 
increases  ? 

26.  If  the  base  be  positive  and  smaller  than  unity  what  is 
the  logarithm  of  a  number  smaller  than  unity?  of  one  larger 
than  unity  ?  How  does  the  logarithm  change  as  the  number 
increases  ? 

27.  What  is  the  logarithm  of  the  product  of  two  numbers  ? 
of  the  quotient  of  two  numbers  ?  of  the  reci2)rocal  of  a 
number? 

28.  What  relation  has  the  logarithm  of  a  power  of  a  number 
to  that  of  the  number  ?  the  logarithm  of  a  root  ? 

29.  What  relation  have  the  logarithms  of  two  numbers, 
each  taken  to  the  other  as  base  ? 

30.  What  relation  have  the  natural  and  the  common 
loorarithms  of  the  same  number? 


Give  the  general  rule,  with  reasons  and  illustrations,  for  : 

31.  Taking   out   a   logarithm    from    the   table,   when    the 
number  is  found  in  the  table  ;  when  not  so  found. 

32.  Taking  out  a  number,   from  its  logarithm,  when  the 
losrarithm  is  found  in  the  table  :  when  not  so  found. 


33.  Define  the  characteristic  and  the  maiitis^a  of  a  loga- 
rithm, and  show  what  relation  the  characteristic  has  to  the 
position  of  the  decimal  point  in  the  number. 


248     PERMUTATIONS,  COMBINATIONS,  PROBABILITIES.     [IX, 

IX.    PERMUTATIONS,   COMBINATIONS,  AND 
PROBABILITIES. 


The  different  groups  that  can  be  made  of  two  or  more 
things,  without  regard  to  order,  are  their  comhinations;  the 
different  groups  that  can  be  made  of  them,  their  order  being 
considered,  are  their  j^ermutations. 

Two  permutations  are  different  when  either  the  things  them- 
selves are  different  or  their  order  of  arrangement  is  different; 
but  two  combinations  are  different  only  when  at  least  one  of 
the  things  contained  in  one  of  them  is  not  found  in  the  otlier. 
E.g.,    ai,  ha,     ac,  ca,     be,  cb,    are  the  six  permutations  of 

the  three  letters     a,  b,  c,    taken  two  at  a  time ; 
but     ab,  ba   are  the  same  combination,    etc,  ca   are  the  same, 

and     be,  cb    are  the  same, 
and  in  all,  there  are  but  three  combinations. 
So,    abc,  bac,  eicb,  cab,  bca,  cba    are  the  six  permutations  of 

the  three  letters     a,  b,  c,    taken  all  together; 
but  there  is  only  one  combination. 
So,  of  four  letters,     a,  b,  c,  d,     taken  three  at  a  time,  there 

are  four  combinations,     abc,  abd,  acd,  bed; 
and  of  each  of  them  can  be  made  six  permutations,  as  above. 
Nearly  all  investigations  as  to  permutations  and  combina- 
tions depend  upon  the  following  self-evident  principle: 

Ax.  If  there  be  a  group  of  m  things  and  a  group  of  n  things 
so  related  that  either  of  the  m  things  may  be  taken  at  random, 
and  then  either  of  the  n  things  may  be  joined  to  it,  there  are 
mn  tuays  in  which  such  pairs  may  be  made  up,  and  no  more. 
E.g.,  if  a  boy  have  5  apples  arid  6  peaches,  he  may  form  30 

different  pairs  of  them,  an  apple  and  a  peach; 
for  the  first  apple  may  go  with  either  of  the  6  peaches,  and  so 

with  the  others. 
So,  if  5  men  enter  a  room  with  G  chairs  the  first  man  has  choice 

of  6  chairs,  the  second  man  of  the  other  5  chairs,  the 

third  man  of  the  other  4  chairs,  and  so  on; 


AX.]  PERMUTATIONS   AND  COMBINATIONS.  249 

and,  while  the  first  man  can  be  seated  in  but  six  ways,  the 
first  two  men  can  be  seated  in  6  •  5  ways,  the  first  three 
men  in  6  •  5  •  4  ways,  and  so  on. 

QUESTIONS. 

1.  Which  of  the  groups  below  are  permutations  and  which 
are  combinations  ? 

the  three-figure  numbers  that  can  be  made  from  n  figures, 
the  products  of  four  factors,  taken  from  ten  given  factors, 
the  parties  of  four  that  can  be  made  up  from  six  men, 
the  ways  twelve  men  can  stand  in  a  row,  or  in  a  ring, 
the  ways  of  dividing  six  things  among  three  men. 

2.  What  are  the  three  combinations,  in  pairs,  of     a,  b,  c"^ 
Make  two  permutations  of  each  of  these  combinations. 

So,  of  each  of  the  combinations  ahc,  abd,  acd,  bed  make 
six  permutations,  thus  forming  the  twenty-four  permutations 
of    «,  b,  c,  d,    taken  in  groups  of  three. 

3.  If  permutations  are  to  be  made  of  the  letters  a,  b,  c,  d, 
how  many  choices  are  there  for  the  first  place  ? 

Make  all  possible  permutations  of  two  by  annexing  to  each 
of  these  first  letters  the  other  three  letters  in  turn:  how  does 
the  number  of  choices  compare  with  that  for  the  first  place  ? 

To  each  of  the  couplets  annex  the  two  remaining  letters 
in  turn,  and  form  all  the  possible  groups  of  three:  how  many 
letters  remain  to  annex  to  each  group  of  three? 

4.  How  does  the  axiom  apply  in  finding  the  number  of 
couplets?  in  finding  the  number  of  threes? 

5.  Of  the  letters  of  the  word  thing,  make  all  the  possible 
permutations  of  two  letters;  of  three  letters;  of  four  letters. 

6.  Why  can  more  permutations  be  made  with  the  same 
letters  arranged  in  a  row  than  in  a  ring  ? 

7.  Find  the  sum  of  all  the  four-figure  numbers  that  can  be 
expressed  by  the  figures  1,  2,  3,  4. 

If  all  these  numbers  be  written  one  under  another,  how 
many  times  is  each  figure  found  in  each  column  ? 

Show  that  the  sum  of  the  numbers  formed  as  above  is 
divisible  by  the  sum  of  the  four  figures  involved. 


250  PERMUTATIONS,  COMBINATIONS,  PROBABILITIES.  [IXThe. 

§  1.  PERMUTATIONS. 

Theor.  1.   The  number  of  ijoninitation.s  of  n  tilings,  all  dif- 
ferent, taken  r  at  a  time,  is     n{7i  —  l)(n  —  2)  •  •  •  (n  —  r-hl). 
For  of  71  things  taken  singly,  there  are  7i  choices  and  no  more; 
i.e.,  Fj^?i  =  u; 

BO,  if  each  of  the  oi  things  be  followed  in  turn  by  each  of  the 
n  —  1  things  that  remain,  there  are  formed   7t(n  —  \} 
couplets,  all  different, 
i.e.,  V2n  =  7i(n  —  1); 

so,  if  each  of  these  n{n—l)  couplets  be  followed  in  turn  by 
each  of  the  7i~2  things  that  remain,  there  are  formed 
n{n~l){n  —  2)  threes,  all  different, 
i.e.,  v.^n  =  7i{?i  —  l){n  —  2); 
and  so  for  the  groups  of  four,  of  five,-  •  -of  ?% 
i.e.,  Pr^i  =  7i{7i  —  l){7i'-2)' '  '(n  —  r-\-\).  Q.E.D. 

Cor.  1.  The  number  of  per7nutations  of  7i  things  all  differ' 
ent,  taken  all  together,  is  the  pi^oduct  of  all  the  integers  from 
1  to  71,  inclusive. 

For  •/  hero  r  is  n,  and  n  —  r-\-\  is  1, 
.-.  v^n  =  n{n-l)(n-2)'  "2,'2'l. 
This  product  is  indicated  by  the  symbol  \n  or  //! ,  and  it  is 
read  factorial  n. 

Cor.  2.  If  in  each  group  of  r  things  soine,  or  all,  7nag  In' 
alike  (permutations  with  repetition),  then  the  nwmber  of  such 
permutations  is  w^ 

For  •.•  there  is  a  choice  of  n  things  for  the  first  place,  tlien  of 
n  things  for  the  second  place,  and  so  on, 
,\  Vr, rep.  u-n-n-n"  -r  times  =  n''. 
Theor.  2.    Tlie  7mmber  of  permutations  ofn  things,  taken 
all  together,  when  p  things  are  alike,  q  things  alike,  r  things 
alike,  and  so  on,  is     n\//)l  q\  r\-  •  - 

For -.-^j!  permutations  of  the  )i  things  are  formed  by  inter- 
changing any;;  things  among  themselves,  while  the 
other  things  stand  fast,  if  the  p  things  be  all  different, 
but  only  one  permutation  if  they  be  alike, 


1,  2,  §  1  ]  PEKMUTATIOXS.  251 

/.  the  whole  number  of  permutations  is  pi  times  larger  if 
they;  things  be  all  diffei-ent  than  if  they  be  alike; 
and  •.•  P,j«  =  III  it  the  n  things  be  all  different,         [tli.  1  cr.  1. 

.*.  the  number  of  permutations  of  ?i  things, y;  alike,  is  n  l/pl 
So,  if  ^  things  be  alike,  r  things  alike,  and  so  on, 

•*•  Pn.  p  alike,  q  alike,  r  alike  .  .  .  ^^  =  ^^  -//^ .  CJ  .)'.-'  "  Q.  E.  D. 

QUESTIOXS. 

1.  In  making  up  permutations,  why  are  there  fewer  choices 
for  each  successive  place  to  be  filled?  at  what  rate  does  the 
number  of  such  choices  decrease? 

What  relation  lias  the  number  of  places  filled,  at  anv  stage 
of  the  process,  to  the  number  of  choices  for  the  next  place? 

2.  Why  is  the  number  of  permutations  of  n  things  taken 
n  —  1  at  a  time  the  same  as  the  number  taken  ')i  at  a  time  ? 

3.  Find  the  number  of  permutations  of  10  things,  all 
different,  3  at  a  time;  5  at  a  time;  7  at  a  time;  all  together. 

4.  How  many  permutations,  3  letters  at  a  time,  can  be  made 
up  from  the  word  mucilage?  from  the  \\OYd j'or?ni(Ia? 

5.  Of  how  many  unlike  things,  taken  all  together,  are  there 
720  permutations"?  5040?  40320? 

6.  What  is  the  value  of  ?/!/(?? -1)1?  of  nl/(n-2)\? 
of  ?i!/(;i-3)I?     of  nl/{u~r)\?     of  nl/rl 

7.  The  number  of  permutations  of  ??  things,  r  at  a  time. 
plus  r  times  their  number  r  —  1  at  a  time,  is 

7i{n-l)-  '  '(?i-r  +  2){n  +  l),   i.e.:  P^Ji  + /■•  P^_i/^  =:P^(;^ +  1). 

8.  In  how  many  ways  can  8  men  stand  in  a  row  ?  n  men  ? 
In  how  many  ways  can  8  men  sit  .around  a  table?  7i  men? 

9.  In  how  many  ways  can  5  prizes  be  given  to  5  boys  ? 

10.  A  man  has  three  ways  of  going  to  his  place  of  business; 
in  how  many  different  ways  can  he  plan  his  route  for  six  days? 

11.  Find  the  number  of  permutations  of  ten  flags  if  three 
be  red  and  seven  blue  ?  if  two  be  red,  three  white,  five  blue? 

12.  How  many  permutations  can  be  formed  fi-om  the  word 
London?   Washington?  Mississippi?  Consfanfinople  ? 


252  PERMUTATIONS,  COMBINATIONS,  PROBABILITIES.  [IX,  Th 

§2.    COMBINATIONS, 

Theor-  3.  The  mimher  of  combinations  of  n  things  all 
different  taken  r  at  a  time,  is  the  quotient  of  the  number  of 
such  'permutations  divided  by  r\ 

For  •/  any  combination  of  r  different  things  gives  r!  permu- 
tations of  '/•  things,  [th,  1  cr.  1. 
.*.  0^?J  =  PrV^*!                                                         QoE.D.  [th.  1. 

OoR.  1.  Tlie  number  of  combinations  of  n  things  taken  n  —  r 
at  a  time,  is  the  same  as  their  number  taken  r  at  a  time. 
For  each  group  of  r  things  leaves  a  group  of  n  —  r  things 

Cor.  2.    The  number  of  combinations  of  n  things  is  greatest 
tohen  r  is  the  integer  nearest  in  value  to  ^n, 
„  n  n  n  —  \  n  7i  —  l  n  —  2 

n  71  —  1       n  —  r-\-l                 n       n  —  r+l    n  —  r 
c..«  =  ^-^ ;^' «-.»  =  ! i ^^1' 

.%  C^?i  is  greatest  when     {n  —  r-V\)/r>\>{n-r)/{r-{-\)y 
i.e.,     when   ^{n  +  \)>r>\(n  —  \),    if  ?i  be  even;    q.e.d. 
and     GrU,  Gr+iU     are  equal  to  each  other,  and  greater  than 

the  other  terms  of  the  series  when 

(n-r  +  l)/r>l^{n-j')/(ri-l). 
I.e.,      when     r  =  i(n-l),     if  ?^  be  oddo  q.e.d. 

Cor.  3.  If  in  each  group  of  r  things  some,  or  all,  mag  be 
alike  (combinations  with  repetitions)  then  the  number  of  such 
combinations  of  n  things  is     n(n  -\-\){n-{-2)-  -  -{n-hr-  l)/r\ 
For,  let  a,b,  c,-  -  -l,  m,  be  any  ?^  +  l  different  things,  form 

all  possible  groups  of  two,     (//  + 1 )  •  n/)l !     in  all, 
and  replace  am  by  aa,     bm  by  bb^     cm  by  cc,     •  •  •     Im  by  ll\ 
then  m  vanishes,  and  there  result   n{n  +  \)/?A    combinations, 

with  repetitions,  of  the  n  things  a,  b,  c,-  '  'J. 
So,  let  a,  h,  c,  "  '  I,  m,  n  be  any  w  +  2  different  things,  form 

all  possible  groups  of  three,    n[n-\-\)[n  +  2)/'d\     in  all, 


3,  §2]  COMBINATIONS.  253 

and  reinlsice  amti  hy  aaay  lm?i  hy  bbb,  •  •  »^  hnn  hy  III, 
abm  by  aba,  acm  by  aca,  •  •  •,  Ikm  by  Ikl, 
aba  by  abb,      acn   by  ace,      -  -  >,     Ikn  by  Jkh] 

then  m,  ti  vanish,  and  there  result    ?i(M  +  l)(?i  +  2)/3!     com- 
binations, with  repetitions,  of  the  n  things,  a,  b,c,-  -  -l, 

ind  so  for  groups  of  four  things,  of  five  things,  «  •  •  of  r  things. 

QUESTIOXS. 

1.  How  many  things  are  taken  at  a  time,  if  the  number  of 
permutations  and  of  combinations  be  the  same? 

2.  Take  the  letters  a,  b,  c,  d,  e  and  join  to  each  of  them 
every  letter  that  follows  it  in  the  list,  thus  making  all  the 
groups  of  two;  form  the  threes  by  joining  to  each  couplet 
every  letter  that  follows  all  its  elements;  so,  the  fours;  the  fives, 

3.  How  many  triangles  can  be  formed  by  joining  three  ver- 
tices of  a  poh'gon  of  n  sides?  how  many  pentagons  by  joining 
five  vertices  ?    AVith  six  points  construct  the  fifteen  pentagons. 

4=  If  an  indefinite  line  be  cut  at  four  points,  how  many  seg- 
ments are  formed  ?  at  six  points  ?  at  n  points  ? 

5.  Show  that  the  formula  for  the  number  of  combinations 
of  n  things  taken  ?•  at  a  time,  may  be  written    7i\/i'\{)i  —  r)\ 

Hence  prove  that     C;,?i  =  C„_^?i. 

6.  Find  the  number  of  combinations  of  10  things  all  differ- 
ent taken  3  at  a  time;  5  at  a  time;  7  at  a  time. 

The  number  of  groups  3  at  a  time  is  the  same  as  their  num- 
ber 7  at  a  time;  and  their  number  5  at  a  time  is  greatest  of  all. 

7.  If  there  be  four  straight  lines  in  a  plane,  no  two  parallel 
and  no  three  meeting  in  a  point,  how  many  triangles  are 
formed  r  if  five  lines  ?  if  7i  lines  ? 

8.  If  there  be  six  points  in  a  plane,  no  three  colinear,  and 
lines  join  them  so  as  to  form  the  greatest  possible  number  of 
figures  of  the  same  kind,  what  will  those  figures  be? 

9.  If  there  be  seven  points  in  a  plane,  and  they  be  joined  so 
as  to  make  triangles  and  quadrangles,  of  which  sort  is  there 
the  greater  number  ?     Draw  the  quadrangles. 

10.  Apply  cor.  3  to  find  the  number  of  terms  in     {a  +  b)\ 


254  PERMUTATIONS,  COMBINATIONS,  PROBABILITIES.  LIX.Th 

Theor.  4.  If  there  he  n  things  all  different,  and  if  p^  q,  be 
any  positive  integers  ivhose  s^im  is  n,  then  there  are  n\/p\q\ 
ways  in  which  these  n  things  can  be  made  np  into  sets  ofp 
things  and  sets  of  q  things. 

For  a  group  of  q  things  is  left  for  every  group  of  p  things  taken, 
and     Q^n^n\lp\{n-p)\^n\/p\q\  Q.E.D.  [th.  3  cr.  1. 

Cor.  1.  If  n=p  +  q  +  r,  the  number  of  sets  of  j)  tilings,  q 
thin{/s,  and  r  things,  is     n\/plq\rl  [above. 

For  •.•  n  things  give     nl/pliq  +  r)]     sets  of  p  things  and  the 

same  number  of  sets  of     q  +  r     things, 
and  •.•  any  sot  of     q  +  r     things  gives     {q -\-r)\/q\r\     sets  of 
q  things  and  tlie  same  number  of  sets  of  r  things, 
.*.  the  whole  number  of  sets  is 

n\/pl{q-\-r)l  x  {q  +  r)\/q\rl  i.e.,  nl/plq\rl        q.e.d. 
So,  when  n  is  the  sum  of  more  than  three  integers. 
E.g.,    if  12  recruits  be  divided  into  squads  of  three,  four,  and 
five,  the  number  of  such  squads  is  12!/3!4!5!; 
if  divided  into  three  equal  squads  and  sent  to  different 

companies,  their  number  is  12!/(4!)l 
it  simply  divided  into  three  equal  squads,  12!/3!(4!)l 

Cor,  2,    The  ntimber  of  combinattons  of  n  different  things 

%n  sets  of  p  things,  q  things,  and  so  on,  luhen    n-p  +  q-\ , 

equals  the  mmber  of  permutations  of  n  things  taken  all  to- 
gether, when p  things  are  alike,  q  tilings  alike,  and  so  on. 
For  the  p\  permutations  oi p  different  things  are  but  a  snigle 

combmation, 
and,  if  the  p  things  be  like  things,  but  a  single  permutation. 

Cor.  3.  The  value  of  the  qnotient  nl/plq\rl-  •  •  is  greatest 
when  no  one  of  the  numbers  p,  Q^r,  -  --  exceeds  any  other  of 
them  by  more  than  a  unit. 

For,  let    p^q-^'l;  Diyp- 

thenv/^!^!  =  (7-t-2);^/!----(7  +  l)K^  +  2)'7^>('7  +  l)K7  +  l)i. 
.%  the  divisor  is  smallest  and  tlie  quotient  largest  when 
p^  q^  r  '  •  '  are  nearest  to  equality. 


4,  §2]  COMBINATIO^iS.  255 

QUESTIONS. 

1.  There  are  as  many  combinations  of  n  things  taken  r  at  a 
time  as  there  are  permutations  of  n  things,  all  taken,  when  r 
things  are  of  one  kind  and  ii  —  r  things  of  another. 

2.  A  cent,  a  dime,  a  quarter  of  a  dollar,  a  half  dollar,  and  a 
dollar  are  each  claimed  by  two  boys:  in  how  many  different 
ways  can  the  coins  be  divided  between  them  ? 

3.  In  the  proof  of  theor.  4,  why  is  the  number  of  combina- 
tions in  sets  of  7;  things  and  q  things  the  same  as  the  number 
when  ;j  things  are  taken  at  a  time  ? 

4.  Prove  theor.  4  cor.  1  when     n  =  p  ^-q^-r  +  s  +  t, 

5.  Whatever  n  may  be,  three  numbers  p,  q,  r  can  be  found 
whose- sum  is  n  and  which  differ  from  each  other  by  not  more 
than  a  unit;  but  if  n  be  a  multiple  of  3  and  p,  q,  r  be  not 
taken  all  equal,  then  the  difference  between  some  two  of  them 
is  at  least  2.     Generalize  this  proposition, 

6.  Show  that  the  number  of  pei-mutations  that  can  be  made 
from  all  of  2ii  things  that  are  of  two  kinds  is  greatest  when 
there  are  n  things  of  each  kind. 

7.  How  can  18  thmgs  be  divided  among  5  persons  in  the 
greatest  number  of  w^ays,  each  person  receiving  the  same  num- 
ber of  things  at  each  distribution  ? 

8.  In  the  expansion  of  (x  +  yY^,  what  term  has  the  greatest 
coefficient?  what  term  of     (^  +  y)^^? 

9.  Of  the  combinations  of  eight  letters  a,  h^  c,-  -  -  ,  taken 
four  at  a  time,  how  many  contain  a  ?  not  a  ?  both  a  and  b? 
a  and  not  h'^  neither  a  nor  b'^  a,  b,  and  c?  «  or  Z*  or  r? 
neither  a  nor  b  nor  c  ? 

10.  In  finding  the  product  (x  i- a)  •  {x -I- a)  -  •  -71  factors,  the 
various  terms  m  the  several  ])urtial  products  are  all  the  possi- 
ble  permutations  of  7i  letters  taken  all  together,  wherein  part 
g,re  r/'s  and  the  rest  afs. 

The  coefficient  of  x^  is  P„^^„aiike>  that  of  x^'^a,    t„)i„_^  aiike'^ 

that  of  X'^'^a^,   Pn^*n-2alike,2alike?    ^ud  SO  OU. 

Hence  prove  the  binomial  theorem. 


256  PERMUTATIONS,  COMBINATIONS,  PROBABILITIES.  [IX,  Th. 

Theor.  5.  If  there  he  n  sets  of  things,  the  first  set  containing 
p  things,  the  next  q  things,  and  so  on,  and  if  combinations  of 
n  things  he  made  iq)  hy  talcing  one  thing  from  each  set,  then 
the  number  of  such  combinations  is  the  product  j^-q-r'  *- 
Foi  each  of  the  p  things  may  be  joined  to  each  of  the  q  things, 
each  of  these  pq  pairs  to  each  of  the  r  things,  and  so  oa 

Q.E.D.         [ax. 

Cor.  I.  The  number  of  combinations  made  by  talcing  h  oj 
the  p  things,  j  of  the  q  things,  k  of  the  r  things,  and  so  on,  is 
the  2)roduct  of  the  number  of  combinations  of  p  things  taken 
h  at  a  time,  of  q  things  taken  j  at  a  time,  and  so  on. 

Cor.  2.  With  the  data  of  cor.  1,  the  member  of  perinuta- 
lions  possible  is    c^p •  Cjq •  c^r -  •  '(h+j  +  k-\ ) ! 

Cor.  3.  The  number  of  possible  combinations  of  some  or  all 
of  p  +  q  +  r-i —  •  things,  of  luhich  p  things  are  alike,  q  things 

alike,  and  so  on,  is    (i?  + 1)  •  (5'  + 1)  •  (>*  + 1) !• 

For  the  jt?  things  may  be  treated  inp  +  l  ways; 
t,e.,  none,  or  one,  or  two,  •  •  •  or^^  of  them  may  be  taken, 
and  each  of  these  (^  +  1)  dispositions  of  the  p  things  may  be 
joined  to  each  of  the  (^'  +  1)  dispositions  of  the  q  things, 
each  of  these  pairs  may  be  joined  to  each  of  the  (r  +  1)  dis- 
positions of  the  r  things,  and  so  on, 
and  the  whole  number  is     ( ;j  + 1)  •  (7  + 1)  •  (r  + 1)  •  •  • ; 
but  •/  this  iuckides  the  case  when  no  thing  is  taken  from  any 
group,  and  this  case  can  not  be  counted, 
.•.c  =  (;^  +  l).(^  +  l)-(r  +  l) 1. 

Cor.  4.   The  number  of  possible  combinations  of  u  differoit 
things,  taken  some  or  all  at  a  time,  is    2"  — 1. 
For  *.•  each  thing  may  be  either  taken  or  left, 
and      either  disposition  of  one  thing  may  be  followed  by  either 
disposition  of  every  other  thing, 
.•,  the  whole  number  of  combinations  including  that  where 
no  thing  is  taken  is     2-2-2- • -^i  times,  =2\       [ax. 
.•.C  =  2~-X,  Q.E-l). 


3'  8  2  ]  COMBINATIONS.  257 

QUESTIONS. 

1.  Write  out  all  the  measures,  prime  and  composite,  of  G; 
of  30;  of  2310;  of  aoc]  of  abed-,  of  a'^-xK 

2.  Out  of  12  democrats  and  16  republicans  how  many  com- 
mittees can  be  made  up,  each  consisting  of  3  democrats  and  4 
republicans?  how  many  committees  of  seven  can  be  made  up 
with  the  condition  that  each  committee  shall  contain  at  least 
two  men  of  each  party? 

3.  How  many  different  signals  can  be  made  by  hoisting  6 
differently  colored  flags  one  above  another,  when  any  number 
of  them  may  be  raised  at  once? 

4.  l.c  GybyC,'  * '  be  n  different  prime  numbers:  find  the  num- 
ber of  different  measures  of  the  product     «" •  h'^-'^-c^-^. . . 

5.  If  all  groups  of  letters  were  words,  how  many  words 
composed  of  two  consonants  and  one  vowel  could  be  made 
from  our  alphabet  of  five  vowels  and  twenty-one  consonants  ? 

6.  The  number  of  permutations  of  n  things  of  two  kinds 
taken  n  at  a  time,  when  some  or  all  are  alike,  is  the  same  as 
the  number  of  combinations  of  n  different  things,  some  or  all 
at  a  time;  hence  the  sum  of  the  coefficients  of  (r/  +  .r)"  is  3". 

7.  From  six  apples,  five  pears,  and  four  plums,  how  many 
selections  of  an  apple,  a  pear,  and  a  plum  can  be  made  ? 

8.  From  a  party  of  six  ladies  and  seven  gentlemen,  how 
many  groups  each  of  four  ladies  and  four  gentlemen  can  be 
formed  ?  how  many  sets  of  four  couples  for  a  quadrille  ? 

9.  Given  m  things  of  one  kind  and  n  things  of  another,  find 
how  many  permutations  can  be  made  by  taking  r  things  of  the 
first  set  and  s  things  of  the  second. 

10.  How  many  different  sums  of  money  can  be  formed  from 
a  cent,  a  half-dime,  a  dime,  three  half-dollars,  five  dollars? 

11.  A  signal  staff  has  five  arms,  each  of  which  may  assume 
four  positions:  how  many  signals  can  be  made? 

12.  If  of  p-hq  +  r  things,  ;?  things  be  alike, // thinirs  alike, 
and  the  rest  all  different,  the  whole  number  of  combinations 
possible  is     (;?  + 1)  •  (g  + 1)  •  2''  - 1. 


258   PERMUTATIONS,  COMBINATIONS,  PROBABILlTIEis.      [IX. 

§  3,   PROBABILITIES. 

The  theory  of  probabilities  is  concerned  with  classes  of 

things  about  whose  individuals  there  is  uncertainty;  it  might 

well  be  called  the  doctrine  of  averages.    It  seeks  to  show  what 

likelihood  there  is  that  a  particular  event  may  liappen  or  fail, 

the  basis  of  computation  being  such  known  facts  as  these: 
that  the  event  considered  must  happen  once,  or  a  fixed 

number  of  times,  in  a  given  number  of  possible  cases; 
that  it  cannot  happen  more  than  such  number; 
that  in  the  past  such  events  have  happened  with  such,  or 

such,  a  degree  of  frequency. 

E.g.,  if  the  twenty-six  letters  of  the  alphabet  be  written  singly 
on  cards  of  the  same  size  and  shape,  and  these  cards  be 
thrown  into  a  box; 

then,  in  twenty-six  successive  drawings,  each  letter  will  be 
drawn  out  once,  and  but  once,  and  at  the  first  di-awing 
one  letter  is  as  likely  to  come  out  as  another. 

So,  as  shown  by  the  Institute  of  Actuaries'  tables,  of  100,000 
healthy  boys  of  ten  96,223  have  reached  the  age  twenty, 
and  609  have  died  between  twenty  and  twenty-one; 
1460  have  reached  ninety,  and  408  have  died  within  a 
year  thereafter;  and  what  has  happened  in  the  past 
may  be  expected  in  the  future  with  ratios  very  slightly 
changed. 
The  yrohaMUty  of  an  event  is  the  ratio  of  the  number  of 

cases  in  which  the  eveut  happens, /<'^i'0/7^Z>/e  cases,ioW\^  wliole 

number  of  cases  considered. 

E.g.,  in  the  example  above,  the  probability  that  the  letter  A  be 
drawn  out,  at  the  first  drawing,  is  the  ratio  1/26;  tliat 
A  be  not  drawn,  25/26;  that  one  of  the  five  vowels  be 
so  drawn,  5/26;  that  neither  of  the  vowels  be  drawn, 
21/26;  that  one  of  the  consonants  be  drawn,  21/26. 

So,  barring  special  conditions  of  health  and  occupation,  the 
probability  that  a  certain  boy  of  ten  live  to  the  age 
of  twenty  is  .96223;  to  the  age  of  ninety,  .0146;  that 
he  die  before  twenty.,  -03777;  before  ninety.  .9854. 


|3]  PROBABILITIES.  259 

QUESTIONS. 

1.  If  the  twenty-six  letters  of  the  alphabet  be  written  on 
separate  cards,  what  is  the  probability 

that  X  be  first  drawn  ?  that  x  be  not  drawn  ? 
that  either  x  or  y  be  drawn  ?  that  neither  x  nor  y  be  drawn  ? 
that  either  x  or  y  or  z  be  drawn  ?  that  neither  x  nor  y  nor 
z  be  drawn  ? 

2.  How  many  different  pairs  of  letters  are  there?        [th.  3. 
If  two  letters  be  drawn  at  a  time  w^hat  is  the  probability  of 

drawing  a  particular  pair?  of  not  drawing  thnt  pair  ? 

3.  How  many  ways  are  there  of  drawing  two  letters  in  suc- 
cession? three  letters ?  four  letters?  [th.  1. 

What  is  the  probability  of  drawing  first  a,  then  b?a,  b 
without  regard  to  order?  A,  then  b,  then  c?  A,  b,  c  without 
regard  to  order?  a,  then  b,  then  c,  then  d  ?  A,  b,  c,  d  with- 
out regard  to  order? 

4.  If  of  men  of  A's  age  one  in  eight  die  in  five  years  there- 
after, and  of  men  five  years  older  one  in  seven  die,  what  is  the 
probability  that  A  will  die  within  five  years?  that  he  will  live 
at  least  five  years?  that  he  will  die  within  ten  years?  that  he 
will  live  ten  years?  that  he  will  die  within  fifteen  years? 

5.  If  of  588  men  of  sixty,  IT,  on  an  average,  die  in  a  year,  of 
the  571  men  of  sixty-one  18  die;  of  the  553  men  of  sixty-two 
19  die;  of  the  534  men  of  sixty-three  20  die;  and  of  the  514 
men  of  sixty-four  21  die,  w^hat  is  the  probability  that  a  man  of 
sixty  lives  one  year?  two  years?  three  years?  four  years  ?  five 
years?  that  a  man  of  sixty-one  dies  within  a  year  ?  two  years? 
three  years?  four  years?  that  a  man  of  sixty-two  lives  till  he 
is  sixty-five?  that  he  dies  before  he  is  sixty-five  ? 

6.  In  throwing  a  single  die  what  is  the  probability  of  a 
six  ?  not  a  six  ?  a  six  or  an  ace  ?  neither  a  six  nor  an  ace  ? 

7.  If  a  boy  with  three  red  marbles  in  his  pocket,  five  white, 
and  seven  blue  ones,  take  out  one  at  random,  what  is  the 
probability  that  it  is  blue  ?  not  blue  ?  a  particular  blue  one? 
not  that  blue  one  ?  red ?  white?  either  red  or  white? 


260  PERMUTATIONS,  COMBINATIONS,  PROBABILITIES.   [IX,  Th. 
SIMPLE    PROBABILITIES. 

If  the  probability  of  a  single  eveut  be  considered,  it  is  the 
swi2:)le  i^rohability  of  the  event;  and  the  sum  of  the  probabil- 
ities for  and  against  such  event  is  always  unity,  i.e.,  certainty. 
E.g.,  that  the  letter  A  be  drawn  out  at  the  first  drawing  the 

probability  is  1/26,  that  it  be  not  drawn  25/26, 
and     1/26  +  25/26  =  1. 

Theor.  6.     If  there  be  tivo  or  more  events  that  are  mutu- 
ally exclusive,  the  prolability  that  some  one  of  them  shall 
happe^i  is  the  sum  of  their  separate  jri' oh  ah  ilities. 
E.g.,  the  probability  that  some  one  of  the  five  vowels  shall  be 
drawn  is  five-fold  that  of  one  of  them. 
The  probability  of  an  event  is  not  the  same  to  every  man; 
to  each  one  it  depends  on  his  knowledge  of  the  facts. 
E.g.,  To  one  who  knows,  of  a  horse,  only  that  he  has  been  a 
frequent  winner,  the  probability  of  his  winning  at  a 
race  to-day  may  be  high; 
but  to  the  groom,  who  knows  that  he  has  gone  lame,  his  de- 
feat is  almost  certaiuo 

PROBABLE    VALUES. 

The  prohahle  value  of  a  sum  of  money  payable  at  some 
future  time  on  conditions  whose  fulfilment  is  uncertain  is 
the  product  of  the  sum  due  by  the  probability  of  receiving  it. 

This  principle  is  of  special  importance  in  life-insurance. 
E.g.,  to  find  the  cost  of  insuring  a  man  of  twenty  for  a  year: 
Were  death  during  the  year  ceitain  aTid  payment  made  at  its 
end,  the  premium  would  be  the  present  worth  for  a 
year,  at  an  agreed  rate  of  interest,  of  the  face  of  the 
policy;  and  at  four  per  cent  the  premium  on  $1000 
would  be  JJS961.54; 
but'.'on  an  average,  609    men   out   of   96223  die  between 
twenty  and  twenty-one, 
,%  the  probability  of  death,  and  of  the  consequent  pay- 
ment of  the  money,  is  609/96223, 
and  8961.54x009/90323  =  $6,086,  the  net  cost  for  $1000. 


6,  §  3  ]  PROBABILITIES.  261 

QUESTIONS. 

1.  If  each  letter  of  the  alphabet  when  drawn  out  be  replaced 
before  another  drawing,  how  many  possible  ways  are  tliere  of 
drawing  two  letters  in  succession  ?  three  letters  ?  the  same 
letter  twice  ?  three  times  ?  [th.  1  cr.  2. 

2.  With  two  dice  what  is  the  probability  of  throwing  a  four 
and  a  three  ?  a  double  four  ?  a  three,  a  four,  and  a  five,  with 
three  dice?  three  fcurs  ? 

3.  What  is  the  probability  of  throwing  exactly  10  in  a 
single  throw  with  three  dice?  12  ?  15?  18?  20?  less  than  8? 

Are  the  probabilities  the  same  to  a  bystander  as  to  a  player 
wdio  has  honest  dice  ?  loaded  dice  ? 

4.  If  a  man  know  that  he  is  to  receive  a  sum  of  money  that 
is  expressed  in  dollars  by  a  three-figure  number  made  up  of 
the  digits  3,  5,  7,  but  know  not  the  order  of  the  digits,  what 
is  the  value  of  his  expectation  ? 

5.  A  fi-iend  is  one  of  tw^o  hundred  passengers  on  a  ship  that 
cariies  a  crew  of  a  hundred  men,  and  it  is  reported  that  one 
man  was  lost  during  the  voyage;  what  is  the  probability,  to  me, 
that  it  was  my  friend  ?  later  it  is  reported  that  it  was  a  pas- 
senger; what  is  the  i3robability  now?  and  stilllater  the  name 
Johnson  is  given,  my  friend's  name,  but  there  were  three 
Johnsons  aboard;  v/hat  now?  what,  to  the  ship's  surgeon  ? 

G.  If  of  1000  boys  of  ten  956  live  to  be  twenty-one,  what 
is  the  present  value  of  $10,000  to  be  paid  on  his  twenty  first 
bii-thday  to  a  boy  now  ten,  the  amount  of  -SI  at  compound 
interest  for  eleven  years,  at  four  per  cent,  being  81.53945? 

So,  if  the  money  earn  five  per  cent,  the  amount  of  $1  for 
eleven  years  being  81.71034  ? 

So,  if  it  earn  but  three  per  cent,  the  amount  being  $1.38423  ? 

7.  If  2)  stand  for  one  payment,  r  for  the  rate  of  interest, 
hf  h>  h,  hy  for  the  probabilities  of  living  one,  two,  three,  four 
years;  find  v,  the  jiresent  value  of  an  annuity  to  run  four 
years,  or  till  pievious  death. 

Make  the  problem  general  by  writing  7i  years,  and  ^,,  l^,  -  ■  ■  l„ 
for  the  probabilities  of  living  one,  two,  •  •  •,  w  yeara 


262   PERMUTATIONS,  COMBINATIONS,  PROBABILITIES.  LiXTa 
JOTXT    PROBABILITIES. 

The  probabilit}^  of  the  simultaneous  occurrence  of  two  or 
more  independent  events  is  their  joi?it  probahility. 
E.ff ,  that  the  letter  A  be  drawn  and  an  ace  be  thrown. 

Theor.  7.  //*  there  he  two  or  more  indefendent  events  sucli 
that  the  simple  probability  of  the  first  is  m/n,  that  of 
the  second,  m' /n' ,  and  so  on,  then  their  joint  probability 
IS  the  product      m/n  •  m' /n'  -  •  • 

For  •.•  the  first  event  happens  m  times  out  of  n,  the  second 

on'  times  out  of  n',  and  so  on,  H'^vp. 

/.  of  any  nn'  joint  events^  mn'  are  favorable  to  the  first 

event, 

and       of  these  mn'  joint  events  favorable  to  the  first  event, 

mm'  are  favorable  to  tlie  second  event. 
I.e.,       of  nn'  joint  events,  mm'  are  favorable  to  both  events. 

.*.  the  joint  probability  of  the  two  events  is  nun' /nn'', 
and  so  if  there  be  three  or  more  events.  q.e.d. 

E.g.,  that  the  letter  A  be  drawn  and  an  ace  be  tlirown,  the 
probability  is  1/2G.1/G, 
that  A  be  drawn  and  an  ace  not  thrown,  1/26 -5/6, 
that  A  be  not  drawn  and  an  ace  be  thrown,  25/26  •!/(), 
that  A  be  not  drawn  and  an  ace  not  thrown,  25/20-5/6; 

and  the  sum  of  all  these  products  is  unity. 

So,  the  probability  that  a  be  first  drawn  and  then  b  is  1/2G'^ 
if  A  be  replaced  after  the  first  drawing, 

and  it  is  1/26-1/25,  i.e.,  1/650,  if  A  be  not  replaced. 

So  the  probability  that  A,  aged  ninety,  and  B,  aged  twenty, 
shall  both  die  witliin  a  year  is  408/1460-609/96223, 
that  both  live  the  year,  1052/1460-95014/96223, 
that  A  lives  and  B  dies,  1052/1460.609/06223, 
that  A  dies  and  B  lives,  408/1460-95614/96223; 

and  the  sum  of  all  these  products  is  unity. 


6,  §3]  PROBABILITIES.  263 

QUESTIONS. 

1.  If  a  bag  hold  three  red  balls,  five  white,  and  seven  blue 
ones,  find  the  probability  of  drawing  three  red  balls  in  suc- 
cession. Show  that  this  probability  is  the  same  as  that  of 
drawing  the  three  red  balls  all  at  once. 

2.  A  bag  holds  7n  white  balls  and  7i  black  ones;  the  proba- 
bility of  drawing  first  a  white  ball  and  then  a  black  one,  and 
so  on  till  all  the  balls  left  are  of  one  color,  is  the  same  as  that 
of  getting  all  the  white  balls  in  a  single  drawing  of  m  balls. 

3.  A  man  on  a  journey  must  make  four  connections  to  get 
through  in  time:  if  the  probability  of  making  each  connection 
be  3/4,  what  is  the  probability  of  making  them  all  ? 

4.  A  man  of  thirty  marries  a  wife  of  twenty- five:  if  of  93 
persons  of  twenty-five  90  reach  thirty,  26  reach  seventy-five, 
and  14  reach  eighty,  what  is  the  probability  of  their  celebrat- 
ing a  golden  wedding  ? 

5.  On  an  average  A  speaks  the  truth  three  times  out  of 
four,  and  B  nine  times  out  of  ten:  what  is  tlie  probability  that 
both  will  assert  a  fact  known  to  them  both  ?  that  both  will 
deny  it?  that  their  statements  will  be  contradictory?  that  one 
or  the  other  of  these  cases  will  occur? 

6.  The  probability  that  A  can  solve  a  certain  problem  is 
2/5,  that  B  can  solve  it  2/3:  if  both  try  it,  what  is  the  prob- 
ability of  its  being  solved?  what,  that  A  succeeds  and  B 
fails?  that  A  fails  and  B  succeeds?  that  both  succeed? 

7.  In  one  purse  are  ten  coins,  a  sovereign  and  nine  shillings; 
in  another  purse  are  ten  coins  all  shillings;  nine  coins  are 
taken  from  the  first  purse  and  put  in  the  other,  then  nine  coins 
are  taken  from  the  second  purse  and  put  in  the  first:  what  is 
the  probability  that  the  sovereign  is  still  in  the  first  purse  ? 

8.  If  the  probability  that  a  ship  will  not  meet  a  gale  be  3/4; 
that,  if  it  meet  one,  it  will  not  be  disabled,  5/G;  that,  if  dis- 
abled, it  will  be  kept  afloat  by  the  pumps,  1/2;  that,  if  the 
pumps  fail,  the  passengers  will  all  escape  by  the  boats,  1/3: 
find  the  probability  of  loss  of  life  by  shipwreck. 


264  PERMUTATIONS,  COMBINATIONS,  PROBABILITIES.  [IX,  Ta 

CoK.  1,  If  p  he  the  simjile  prohahility  of  the  occurrence  of 
an  event  in  one  trial,  then  p^  is  the  prohahility  that  it  occurs 
i7i  all  of  n  successive  trials. 

E.g.,  the  probability  that  the  letter  A  be  drawn  twice  in  suc- 
cession, being  replaced  after  the  first  drawing,  is  1/2G^ 

Cor.  2.  If  the  prohaMlity  of  the  occurrence  of  an  event  in 

one  trial  he  p  and  of  its  failure  q,  then  the  prohahiliiy  of  its 

occurrence  r  times  in  n  trials  is  the   {n  —  r  +  l)th   term  of  the 

expansion  of    {p  +  qY, 

For  the  probability  that  the  event  occurs  n  times  in  succession 
is  p'^,  [cr.  1. 

that  it  occurs  n  —  1  times  and  fails  once,  is  the  productjo**"**^ 
taken  as  many  times  as  there  are  permutations  of  n 
things  with  n  —  1  of  them  alike,  i,e.,  n'p^~'^q,      [th.  6. 

that  it  occurs  ?z  — 2  times  and  fails  twice,  is  the  product 
pn-2,gZ  taken  as  many  times  as  there  are  permutations 
of  n  things  with  n  —  2  of  them  alike  and  2  alike, 

I.e.,      in(n  —  1)  -p"-^-  q^,     and  so  on; 

that  it  occurs  r  times  and  fails  n  —  r  times,  is  the  product 
pr.qn-r  takcu  as  many  times  as  there  are  permutations 
of  n  things  with  r  of   them  alike  and   n  —  r  alike, 

i,e.,      n(n-l)"  '{n-r  +  l)/rl'p'''q''-\  q.e.d. 

Cor.  3.  The  prohability  that  an  event  occurs  at  lefts f  r  times 

in  n  trials  is  the  siwi  of  the  first  n  —  r-\-l  terms  in  the  expaii- 

sion  of  (p  +  qY* 

Cor.  4.  That  value  of  r  for  which  the  prohahility  is  greatest 

is  the  largest  integer  i7i  q(n-{-l). 

For  the  expression     n{n  —  l){n-2)" '{n-r  +  l)/rl-p''-^'q^ 

.     .     1       n  —  r-\-\q      ^     n  —  rq       rxi    o       « 

is  greatest  when >  1  >  — — :  •  -,     [th.  3  cr.  2. 

&  r        p  r-\-\  p       ^ 

^.e.,  when     nq-rq  +  q>rp,    and    Nq  +  q>r(p  +  q), 

and  when     rp-\-p>nq-rq,    and    r(p  +  q)>nq~p; 

and  •.•;)  +  v  =  l,  [hyp. 

.•.  it  is  greatest  when     q{n  +  1)  >  r  >  q(n  + 1)  - 1.      Q.  E.  u 


6,  §31  PROBABILITIES.  265 

QUESTIONS. 

1.  If  the  probability  that  a  man  of  fifty  live  to  be  eighty  be 
1/5,  what  is  the  probability  tJiat  of  six  men  of  fifty,  three  at 
least  reach  eighty  ?  four  at  least  ?  five  at  least  ?  all  of  the  six  ? 

2.  If  on  an  average,  of  the  ships  engaged  in  a  certain  trade, 
nine  out  of  ten  return  safely,  find  the  probability  that  out  of 
five  ships  expected  three  come  into  port. 

3.  In  how  many  trials  is  there  an  even  chance  of  throwing 
double  sixes  with  two  dice  ?  a  single  six  with  one  die  ? 

4.  Two  players  A,  B,  of  equal  skill  are  interrupted  in  a  game 
when  A  wants  two  points  of  winning  and  B  three:  show  that 
the  prize  should  be  divided  in  the  ratio  11/5. 

5.  The  probability  that  a  man  will  die  within  a  year  is  1/8; 
that  his  wife  will  die,  1/10;  that  his  son  will  die,  1/60:  if  all 
three,  or  any  two  of  them,  be  living  at  the  end  of  the  year 
they  are  to  receive  810,000  in  equal  shares;  what  is  the  value 
of  the  expectation  of  each  of  them,  interest  at  five  per  cent  ? 

6.  At  simple  interest,  five  per  cent,  find  the  present  value 
of  an  annuity  of  8200  to  run  two  years,  and  contingent  on 
the  joint  lives  of  two  persons  whose  probabilities  of  living  for 
the  next  two  years  are  76/77,  74/75  for  the  first  person,  and 
66/76,  65/75  for  the  other,  the  annuity  being  payable  only  if 
both  be  living;  payable  if  either  be  living. 

7.  Three  men  A,  B,  C,  throw  a  die  alternately  in  the  order 
of  their  names,  and  whoever  first  throws  a  five  wins  8182; 
show  that  their  expectations  are  872,  860,  850. 

8.  It  is  a  question  whether  A  has  been  elected;  B  tells  C 
that  D  told  him  that  A  was  elected,  but  C  thinks  it  an  even 
chance  whether  D  said  elected  or  not  elected.  A  is  elected  if  B 
and  D  both  spoke  truly  or  both  falsely,  find  the  probabilities. 

9.  If  of  thirteen  aldermen  at  dinner  the  probabilities  of 
living  a  year  be  13/14,  14/15,  15/16,  16/17,  17/18,  18/19, 
19/20,  20/21,  21/22,  22/23,  23/24,  24/25,  25/26;  what  is  the 
probability  that  all  of  them  live  a  year?  what,  that  some  one 
of  them  dies  within  a  year  ?  what,  if  there  be  but  twelve  men  ? 


266     PERMUTATIONS,  COMBINATIONS,  PROBABILITIES.     [IX, 

§  3.  QUESTIONS   FOK  REVIEAV. 

Define  and  illustrate: 

1.  Permutations;  permutations  of  n  different  things  taken 
r  things  at  a  time  ;  permutations  with  repetitions. 

2.  Combinations;  combinations  of  n  different  things  taken 
r  things  at  a  time ;  combinations  with  repetitions. 

3.  A  factorial  number. 

4.  The  probability  of  an  event  ;  simple  probability. 

5.  Probable  values  ;  the  joint  probability  of  two  events. 

6.  State  the  fundamental  principle  of  permutations  and 
combinations. 

7.  Show  how  to  write  out  the  permutations  of  n  letters  in 
groups  of  two  letters;  of  three  letters;  •  •  •  of  r  letters. 

8.  Show  how  to  write  out  the  combinations  of  n  letters  in 
groups  of  two  letters;  of  three  letters;  •  •  •  of  r  letters. 

State  and  prove  the  general  rule  for  finding : 

9.  The  number  of  permutations  of  n  things,  all  different, 
in  groups  of  r  different  things; 

all  together; 

in  groups  of  r  things  with  repetitions  allowed. 

10.  The  number  of  permutations  of  n  things,  all  together, 
with  jtj  things  alike,  q  things  alike,  r  tilings  alike,  and  so  on. 

11.  The  number  of  combinations  of  n  things,  all  different, 
in  groups  of  r  different  things  ; 

in  groups  of  r  things  with  repetition  allowed  ; 

in  groups  of  'p  things,  q  things,  ;•  things,  and  so  on. 

12.  The  number  of  combinations,  in  groups: 

of  one  thing  from  each  of  n  sets  of  things  that  contain  p 
things,  q  things,  r  things,  and  so  on; 

of  h  things  out  of  a  set  of  ^;  things,  y  things  out  of  a  set  of 
q  things,  h  things  out  of  a  set  of  r  things,  and  so  on; 

of  some  or  all  of  />  +  </  +  ?•+•••  things,  when  j?>  things  are 
alike,  q  things  alike,  r  things  alike,  and  so  on; 

of  some  or  all  of  n  different  things. 


§4.]  QUESTIONS  FOR  REVIEW.  267 

13.  The  value  of  rtliat  makes  Cyii  the  greatest. 

1-4.  The  relations  of  i^,  q,  r,-  •  •  that  give  the  greatest 
number  of  combinations  of  n  things  in  sets  of  i^  things,  q 
things,  r  things,  and  so  on. 

Prove  that: 

15.  Of  n  different  tilings  there  as  many  combinations  in 
groups  of  n  —  r  things  as  in  groups  of  r  things. 

16.  Of  n  different  things  there  are  as  many  combinations 
in  sets  of  iJ  things,  of  q  things,  of  r  things,  and  so  on,  as 
tliere  are  jiermutations  of  n  things  taken  all  together,  when 
IJ  things  are  alike,  q  things  alike,  r  things  alike,  and  so  on. 

As  deductions  from  the  principles  established  in  this  chapter: 

17.  Prove  the  binomial  theorem. 

18.  Find  the  number  of  terms  in  the  expansion  of  a  power 
of  a  binomial. 

19.  Find  the  term  of  the  expansion  whose  value  is  greatest. 
State  and  prove  the  rule  for  finding  the  probability: 

20.  That  some  one  of  n  mutually  exclusive  events  will 
occur. 

21.  That  two  or  more  events  of  known  probability  will 
occur  jointly. 

22.  That  an  event  will  occur  in  all  of  n  successive  trials. 

23.  That  it  will  occur  exactly  r  times  in  n  trials. 

24.  That  it  will  occur  at  least  r  times  in  n  trials. 


25.  State  and  prove  the  rule  for  finding  the  most  probable 
number  of  successes  in  n  trials  of  things  of  known  proba- 
bility. 

26.  Show  how  far  the  doctrine  of  probabilities  can  be  applied 
in  any  individual  case;  and  where  it  fails. 

27.  Three  works,  one  of  two  volumes,  one  of  three,  and 
one  of  four,  stand  side  by  side:  what  is  the  probability  that 
the  volumes  of  each  work  stand  in  their  proper  order  ? 

28.  If    6;i8=:C;^2l8,    find  C^r, 


268     PERMUTATIONS,  COMBINATIONS,  PROBABILITIES,     ixi. 

29.  Find  the  whole  niiiiiber  of  combinations  of  2)  +  q-\-r 
things  of  which  p  things  me  alike,  q  things  alike,  and  the 
rest  all  different. 

30.  Find  the  odds  against  A's  winning  four  games  before  B 
wins  two  at  a  game  where  A  is  twice  as  good  a  player  as  B. 

31.  If  a,  b,  c,'  '  '  11  be  different  prime  numbers,  what  is  tlie 
number  of  metisures  of  the  product  a^^'h^^-c^^'  •  'l^-m^-n^. 

32.  Find  the  chance  of  throwing  at  least  eight  in  a  single 
throw  with  two  dice;  with  three  dice. 

33.  A  and  B  play  a  set  of  games  in  which  A's  chance  in 
each  game  is  p,  and  B's  q\  show  that  the  probability  of  A's 
winning  m  games  out  of  m  +  n  games  is 

2r'[  +  np-Yn{n-Vl)p^/2\  +"- 

+  ?2(?i  +  l)--.(?i-|-??i-2);y"-V(m-l)!]. 

34.  From  a  bag  that  holds  n  balls  a  man  draws  out  a  ball 
and  replaces  it  n  times:  what  is  the  probability  of  his  having 
drawn  every  ball  in  the  bag? 

35.  Into  a  box  having  three  equal  compartments  four  balls 
are  thrown  at  random:  show  that  there  are  eighty-one  pos- 
sible arrangements;  and  find  the  probability  that  tlie  four 
balls  are  all  in  one  compartment;  that  three  of  them  are  in 
one  compartment  and  one  in  another;  that  two  of  them  are 
in  one  compartment  and  two  in  another;  that  two  of  them 
are  in  one  compartment,  and  one  in  each  of  the  others. 

36.  The  number  of  combinations  of  n  different  things  in 
groups  of  r  tilings,  with  repetition,  is  the  number  of  combi- 
nations of  n  +  r  —  1  things  in  groups  of  r  things  without 
repetition. 

37.  The  number  of  possible  combinations  of  n  things  in 
groups  the  number  of  whose  elements  is  even  differs  by  one 
from  the  number  of  such  combiiuitions  in  groups  the  number 
of  whose  elements  is  odd. 


INDEX. 


Abstract  numbers,  definition  of,  2. 
equal,  4. 

negative,  as  operators,  2,  20. 
product  of,  6, 
sum  of,  22. 
Addition,  definition  and  sign  of,  22. 
associative  and  commutative,  24; 
multiplication  distributive  as  to, 

26. 
of  fractions,  24,  66. 
of  radicals,  172. 
rules  for,  40,  66,  172. 
Algebra,     as     distinguished    from 
arithmetic  and  geometry,  1. 
primary  operations  of,  36-69 
Algebraic  expressions,  36-39. 
Alternation,  proportion  by,  144. 
Anti-logarithm,  definition  of,  232. 

rule  for  finding,  240. 
Arrangement,  46,  58. 

letter  of,  108. 
Associative  operations,  8,  24,  154. 
Axioms,  of  equality,  70,  71. 
of  inequality,  148,  150. 
relating    to     combinations    and 
permutations,  248. 
to  products,  sums,  and  differ- 
ences of  integers,  100. 
of  entire   functions   of    one 
letter,  110. 
Base,  of  logarithm,  233. 
change  of,  234. 
of  power,  30. 
Binomial,  definition  of,  36. 
surds,  170,  172,  174,  192,  194. 
theorem,  162. 
Coefficients,  definition  of,  38. 

use  of  detached,  50,  58. 
Combinations,  definition  of,  248. 


Combinations,    fundamental    prin- 
ciple of,  248. 

maximum  number  of,  252,  254. 

of  11  things  all  different,  252-256. 
some  or  all  alike,  252,  256. 
Commutative  operations,  10,  24,  28, 

110,  158. 
Composition,  proportion  by,  144. 
Constants,  definition  of,  136. 
Continuous   and  discontinuous  va- 
riables, 138. 
Contraction,  in  division,  60. 

in  finding  roots,  188,  190. 

in  multiplication,  54. 
Cross  multiplication,  46. 
Decimal,  value  of  repeating,  222. 
Degree,  of  equation,  70. 

of  term  or  expression,  38. 

of  terms  of  product,  44. 
Detached  coefficients,  their  use, 

in  division,  58. 

in  multiplication,  50. 
Discussion  of  a  problem,  84. 
Division,  definition  and  signs  of ,  16. 

arrangement  of  terms  in,  58. 

checks  in,  58. 

contraction  in,  60. 

of  fractions,  66. 

rules  for,  56-63,  66,  174,  242. 

proportion  by,  146. 

s}Tnmetry  in,  60. 

synthetic,  62. 

use  of  detached  coefficients  in,  58. 

use  of  type-forms  in,  60. 
Elimination,  78. 
Equations,  definition  of,  70. 

dependent,  80. 

elements    of,  known     and    un- 
known, 70. 

269 


270 


INDEX. 


Equations,  exponential,  242. 
indeterminate,  80,  86, 
involving  surds,  178. 
quadratic,  198-217. 

complete  and  incomplete,  198. 
formation  of,  from  roots,  200. 
general  forms  of,  202. 
liigher  equations  solved  as,  204. 
maxima     and    minima    deter- 
mined by,  216. 
of  one  unknown  element,  198. 
properties  of  roots  of,  200. 
simultaneous,  206-213. 
solved  by  factoring,  200. 
special  cases  of,  200. 
reciprocal,  215. 
roots  of,  70. 
simple,  70,  99. 
fewer     conditions     than     un- 
known elements,  86. 
general  forms  of,  76,  92. 
more  conditions  tlian  unknown 

elements,  86. 
of  one  unknown  element,  72. 
of  three    or    more    unknown 

elements,  88,  90. 
of  two  unknown  elements,  78. 
special  problems,  74,  82. 
Euclid's  process  for  finding  highest 

common  measures,  102,  112. 
Evolution,  definition  of,  32. 
an  inverse  process,  180. 
geometric  illustration  of,  188. 
rules  for,  180-186,  190. 
Exponents,  definition  of,  30. 
fraction,  164. 

work  with,  30,  32,  166,  168,  230, 
231. 
Expressions,  definition  of,  36. 

degree  and  kinds  of,  38,  76,  170. 
Factors,  definition  of,  6. 
entire,  118. 
linear,  122. 


Factors,  of  highest  common  meas 
ures,  124. 
of  lowest  common  multiples,  126. 
prime,  106,  116. 
rules  for  finding,  118-122. 
Fraction  powers,  164-169. 
Fractions,  as  exponents,  164-169. 
complex,  66. 

highest    common  measures   and 
lowest  common  multiples  of, 
126,  130,  132. 
operations  on,  14,  24,  66. 
rational  denominator  of,  176. 
reduction  of,  14,  64,  124. 
simple,  2. 
Functions,  definition  of,  108. 
entire,  of  single  letter,  108. 
measures     and    multiples    of, 
108-117. 

Identity,  definition  of,  70. 
Incommensurable  numbers,  defini- 
tion of,  136,  152. 

multiplication  by,  154. 

multiplication  of,  associative  and 
commutative,  154,  156,  158. 

previous  theorems  applied  to,  158. 
Incommensurable  powers,  228-231. 
Induction,  proof  by,  162. 
Inequalities,  4,  148. 

greater-less,  150. 

larg(>r-smaller,  148. 
Insurance,  cost  of,  260. 
Integers,  definition  of,  2. 

measures  and  multiples  of,  100- 
106,  118. 
Integer  powers,  30-32. 
Interpolation  of  terms,  220-223. 
Inversion,  proportion  by,  144. 
Involuticm,  definition  of,  30. 

general    j)rinciples   of,   32,    164, 
174,  234. 

Logarithms,  232-245. 


INDEX. 


271 


Logaritlims,  base  of  system  of,  232. 

change  of,  234. 
characteristic   and   mantissa    of, 

236. 
of   products,  quotients,   powers, 

and  ro-   s,  234,  242. 
properties  of,  to  base  ten,  236. 
systems  of,  234. 
table  of,  244,  245. 

limitations  in  the  use  of,  241. 

rules  for  using,  238-242. 
Measures,  definition  of,  100,  108. 
highest  common.  100,  110. 

composition  of,  106,  117,  124. 

Euclid's  process  for,  102,  112. 

rules  for  finding,  128,  130. 
Monomials,  definition  of,  36, 
Multiples,  definition  of,  100,  108. 
lowest   common,  100,  110. 

composition  of,  126. 

rules  for  finding,  132. 
Multiplication,  definition  and  signs 

of,  6. 
arrangement  of  terms  in,  46. 
associative,  8,  154. 
checks  in,  42. 
commutative,  10,  156. 
contraction  in,  54. 
cross,  46. 

distributive  as  to  addition,  26. 
of  fractions,  66. 

of  powers,  30,  32,  166,  231,  234. 
of  radicals,  174. 
rules  for,  42-54,  66,  174,  242. 
use  of  detached  coefficients  in,  50. 

of  symmetry  in,  52. 

of  type-forms  in,  48. 
Multiplier  always  abstract,  6. 
Numbers,  definition  and  kinds  of, 

2,  4. 
commensurable   and    incommen- 
surable, 136. 
entire,  118. 


Numbers,  expression  of,  4,  20. 

positive  and  negative,  18-21. 

prime  and  comjiosite,  104,    106, 
114. 

prime  to  each    other,   104,    105, 
106,  114,  116. 
Operations,  inverse,  16,  28,  32,  180. 

repetitions  and  partitions,  2. 
Operator,  definition  of,  2. 

office  of  negative,  20. 

position  of,  4. 
Opposites,  26. 
Parentheses,  use  of,  6,  28. 
Partition,  definition  and  sign  of,  4. 
Permutations,  definition  of,  248. 

fundamental  principle  of,  248. 

of  n  things  all  different,  250. 
not  all  different,  250. 
Polynomials,  definition  of,  22,  36. 

degree  of,  38. 

operations  on,  40,  42,  56. 

roots  of,  180-184. 
Positive  and  negative  numbers,  18. 

expression  of,  20. 
Powers,  definition  and  sign  of,  30. 

and  roots,  162-197. 

commensurable,  164. 

fraction,  164-169. 
equal,  166. 

operations  on,  166-169. 
series  of,  164. 

incommensurable,  164,  228-231. 

integer,  30. 

like  and  unlike,  164. 

of  powers,  32,  33,  166,  230. 

operations  with,  230,  231. 

products  and  quotients    of,  30, 
31,  32,  168,  230,  231. 
Prime  numbers,  104,  106,  114, 116. 
Probabilities,  258-265. 
Problems,  discussion  of,  84. 
Product,  definition  of,  6,  154. 

form  of,  44. 


272 


INDEX. 


Product,  logarithm  of,  232,  234. 
of  abstract  numbers,  6. 
of  numbers  expressing  partition 

or  repetition,  14. 
of  polynomials,  26. 
of  powers,  30,  32,  168,  230,  231. 
of  product  of  two  numbers  by  the 

reciprocal  of  one  of  them,  16. 
of  simple  fractions   and  of    re- 
ciprocals, 14. 
Progressions,  see  Series, 
analogies  of  the,  226. 
Proportion,  definition  of,  142. 
continued,  143,  146. 
properties  of,  144,  146. 
transformations  in,  144,  146. 
Proportionals,  142,  144. 
Quadratic  equations,  198-217. 
Quadrinomials,  definition  of,  36. 
Quantity,  definition  of,  2. 
Quotient,  definition  of,  16. 
form  of,  58. 
logarithm  of,  234. 
of  powers,  31,  32, 168,  230-234. 
Radicals,  definition  of,  170. 
kinds  of,  170. 
operations  on,  172-176. 
Ratio,  definition  of,  142. 
direct  and  inverse,  142. 
in  geometric  progression,  222. 
Rationalization,  of  fractions,  176. 

of  an  equation,  178. 
Reciprocals,  definition  of,  14. 

product  of,  14. 
Repetition,  definition  and  sign  of,  4. 
Review  questions,  34,  68,  94,  134, 

160,  196,  214,  246,  266. 
Roots,  of  l)iii()mial  surds,  192,  194. 
of  ei  J  nations,  70. 
of  nnniixTs,  33. 
of  numerals,  186-190. 

by  contraction,  188,  190. 
of  polynomials,  180-184. 


Roots,  square,  geometric  illustra 

tion  of,  188. 
Root-index,  33. 

Series,  definition  of,  37,  218. 
arithmetic,  218-221. 
continuous,  221,  228. 
interpolation  of  terms  in,  220 
finite  and  infinite,  37. 
geometric,  222-225. 
continuous,  225,  228. 
interpolation  of  terms  in,  224. 
harmonic,  226. 

major  and  minor  terms  of,  228. 
Signs,  of  aggregation,  6. 
of  continuance,  8. 
of  equality  and  inequality,  4, 15C 
of  inference,  8. 
of  infinity,  76. 

of  operation,  4,  16,  20,  22,  32. 
of  quality,  20. 

of  repetition  and  partition,  4. 
of  variation,  140. 
radical,  33. 
Subtraction,  theory  of,  28. 
of  fractions,  66. 
rules  for,  40,  66,  172. 
Sum  of  abstract  numbers,  22. 

of  concrete  numbers,  22. 
Surds,  definition  of,  170. 
equations  containing,  178. 
kinds  of,  170,  174,  176. 
properties  of,  174,  192. 
roots  of  binomial,  192,  194. 
Symmetry,  definition  of,  38. 
as  an  aid  in  division,  60. 
in  multiiilicution,  52. 
Synthetic  division,  62. 

Tensors  and  versors,  21. 
Trinomials,  definition  of,  36. 
Type-forms,  their  use,  48,  60,  12C 

Unit,  definition  of,  3. 

Variables  and  variation,  130-140. 


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